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R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

Table of Contents

1.

2.

3.

3.1.

3.2.

3.3.

4.

4.1.

4.2.

4.3.

4.4.

4.5.

5.

5.1.

6.

6.1.

6.2.

6.3.

6.4.

6.5.

7.

7.1.

7.2.

7.3.

7.4.

7.5.

8.

8.1.

References

Appendix A.

Notation

Introduction

The Mutation-Selection Chemostat model

The MSC model generates a dissipative dynamical system

The mutation matrix, M

Comment: mutations are measured per unit time

Trade-offs: theory and evidence

Rate-yield trade-off: definitions

Rate-yield trade-off: evidence

Rate-yield trade-off: biochemical basis

Is the rate-yield trade-off likely to be common?

Rate-affinity trade-off

The bifurcation structure of steady-state solutions

Lethal Mutagenesis

Predictions of the MSC model: the coexistence hypothesis

Steady-state sugar consumption decreases with increasing mutation rate

Cell density scales linearly with S0

Diversity is independent of S0

The MSC model can maintain both fit and flat quasispecies

Extension: asymmetric mutation matrices

An individual-based, stochastic model

Neutrality in the stochastic model

Competitive exclusion in the stochastic model

Co-maintenance of the fit and the flat in the stochastic model

Preventing valley-crossing lineages: mutational barriers

Robustness to changes in mutation kernel

Co-maintenance: a toy model

A toy population genetical model

2

2

3

5

6

8

9

9

9

11

12

14

14

18

19

20

20

21

22

26

28

29

29

30

32

34

34

36

37

39 Model parameters

1

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2R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

1. Notation

We use the following Nature convention: ‘scalars’, meaning real numbers, will be written using roman font

as in the statement u ∈ R, whereas ‘vectors’, meaning members of Rnfor n ≥ 2, will be written in bold,

roman font as in u ∈ R2. Matrices with be capitalised, so that M will represent a matrix that can act on a

vector by multiplication, written as Mu, and I will denote the identity matrix that satisfies Iu = u for all u.

We shall write 0n×nhighlighting the dimension, n×n, of a matrix containing only zero entries. The notation

diag(u) will be used to denote the square, diagonal matrix whose entries along the main diagonal are given

by a vector u.

If F(u,v) is a smooth function (meaning Fr´ echet differentiable) of two vector-valued quantities, F :

Rp× Rq→ Rr, we say that duF(u,v) is the partial derivative of F with respect to the first variable.

This derivative is a linear mapping from Rpto Rrthat can be identified with an r × p matrix. We will

write d2

tion d(u,v)G(u,v,w) will denote the derivative with respect to the first two vector arguments of a function

G(u,v,w).

The notation ?u,v? represents the finite-dimensional, Euclidean inner product, ?u,v? =?n

whenever u = (u1,...,un) and v = (v1,...,vn) are two vectors of the same dimension; to reduce notational

clutter where no confusion arises, column and row vectors will be used interchangeably. So, ?u,u? =?n

?A? = +

and written uv, we interpret this as a pointwise product: uv = (u1v1,u2v2,...,unvn). The uniform vector

(1,1,...,1) will be written with bold font as 1 and so ?1,u? =?n

If u ∈ Rnis a vector and s is a scalar, by u − s we mean the vector u − s1 = (u1− s,u2− s,...,un− s);

this dimensionally incompatible notation for vector-scalar addition is consistent with the numerical package

Matlab. By span{u} we mean the one-dimensional linear space {λ · u : λ ∈ R} and then span{u}⊥= {v ∈

Rn: ?u,v? = 0}. Following convention, the vector u is said to satisfy the inequality u > 0 if both ui≥ 0 for

all i = 1,...,n and there is a j such that uj > 0. We shall write u ≥ 0 if ui≥ 0 for all i and then u ? 0

means that ui> 0 for all such i.

A diversity measure is a functional H : Rn→ R that is scale-invariant, permutation-invariant, maximised

at the uniform vector 1 and minimised at the competitive exclusion state (1,0,...,0), moreover the functional

should be non-constant: H(1,0,...,0) < H(1). Hence, H(λu) = H(u) for all scalars λ > 0 and all non-negative

vectors u > 0 and if P is a permutation matrix then H(Pu) = H(u). Although not often written using this

notation, the following version of Simpson’s index is a diversity measure in this sense:

uuF(u,v),d2

uvF(u,v) and d2

vvF(u,v) to represent second partial derivatives. Analogously, the nota-

i=1uivi, defined

i=1u2

i

and so ?u? = +?u,u?1/2is the Euclidean norm (or length) of u. If A = (aij)n

??n

i,j=1is a matrix, we will write

i,j=1a2

ij

?1/2

for its Frobenius norm. When two vectors of the same dimension are juxtaposed

i=1ui.

HS(u) = −

????

u

?u,1?−1

n1

????

2

.

Diversity measures are not unique, we could add any constant to HSand it would still satisfy our requirements

for being a diversity measure. Indeed, entropy may be used to measure diversity.

2. Introduction

The purpose of this supplement is to present three different theoretical models of microbial growth and evo-

lution in a spatially unstructured environment, each of which can exhibit the phenomenon of co-maintenance

of the fittest and the flattest. The structure shared by all the models is a fitness landscape that determines

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION3

which microbial cells have the highest absolute fitness, a synonym for their growth rates, and whereby the

landscape possesses the following two features.

Firstly, the landscape has both a fit and a flat peak. While the term fittest simply refers to those microbial

cells at the pinnacle of the fitness landscape, the flattest [46] refers to local fitness peaks with a lower absolute

fitness than the fittest but which are more mutationally robust (see supplement fig. 1). Mutational robustness

means that while a mutant offspring cell will suffer a reduction in fitness relative to its parent at the local

peak, the smaller this reduction the more robust the peak is to mutation.

Secondly, the fitness landscape depends on the concentration of a single, limiting carbon source present in

the environment. This carbon source is necessary for the cells to divide and, in doing so, they must consume

some of the carbon and so deplete the quality of the environment. This, in turn, necessarily reduces the

absolute fitnesses of all the cells.

supplement fig. 1 – Fit and flat peaks: assuming a correlation between genotype and

phenotype, ‘flat’ describes a low fitness peak that is more robust to mutations than the fit peak.

phenotype

fitness

fit flat

We show that these two properties can be sufficient to maintain a population of cells at both fit and flat

peaks together in equilibrium, provided mutation rates are neither too high nor too low. As the two peaks

represent microbial cells that express different phenotypes within one population, this provides a framework

to understand when multiple phenotypes may coexist in an unstructured envrionment. We will argue that the

peaked structure found in the fitness landscape may be due to metabolic trade-offs whose existence has been

postulated by others, between growth rate and growth yield for example. However, microbial cells can be the

subject of other trade-offs that may conspire to produce such a coexistence effect, but a detailed discussion of

this phenomenon is beyond the scope of this document and will form the basis of a series of future publications.

3. The Mutation-Selection Chemostat model

Our first mathematical model is the mutation-selection chemostat (MSC) equation, the following ordinary

differential equation

d

dtf

d

dt∆

d

dtS

=

?(M − I)f + G(S)f − ?G(S),f?f,

(1a)

= ∆(−d + ?G(S),f?),

(1b)

=

d(S0− S) − ∆?U(S),f?

(1c)

As regards the use of the term ‘mutation’ in this model, DNA-based mutations may apply, but mutation is

better interpreted as a transition between different heritable states. For example, heritable epigenetic changes,

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4 R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

including transitions between alternative metabolic and regulatory configurations [31, 43] are possible. Thus,

instead of ecotypes or genotypes, we will refer throughout the main manuscript to heritable ‘types’.

The mathematical notation used in the MSC model are the following.

(A1). S, measured in micrograms per millilitre (µg/ml) unless otherwise stated, is the concentration of the

limiting carbon source in the culture vessel of a chemostat at time t hours (h), ∆ is the total density

of cells per ml, S0(µg/ml) is the concentration of the carbon source held in a supply vessel, ? is the

phenomenological mutation rate described in the main text of the manuscript and assumed to be the

same for all types, d (per h) is the washout rate of the chemostat. (The volume of the culture vessel

of the chemostat is not specified.)

(A2). There are n cell types, each type is labelled by a unique index i where i = 1,...,n; the mutational

neighbours of type i are types i + 1 and i − 1, analogous terminology is used at the boundaries 1 and

n. (See supplement figs. 2(a) and 2(b) below for an illustration.)

(A3). The vector f(t) = (f1(t),f2(t),...,fn(t)) contains the frequencies of each type, so that?n

holds for all times t ≥ 0;

(A4). the i-th type has maximal uptake rate Vmax· xi, where Vmaxis a fixed constant measured in units

of µg per cell per unit time and xi= i/n is a dimensionless value between 0 and 1; the latter will be

called the normalised maximal uptake rate throughout.

(A5). The i-th type has a half-saturation constant for resource, K(xi), where K(x) is a smooth function;

K(x) is measured in the same units as S, µg/ml.

(A6). The column vector U(S) = (U(x1,S),U(x2,S),...,U(x2,S)) of uptake rates of all types is determined

by the Monod function U(x,S) = Vmax

(A7). The yield of biomass per unit of resource of type i is ci(in units of cells per µg) and we assume that

cican be derived from the maximal uptake rate of type i. As a result, we shall write ci= c(xi) where

c(·) is a smooth function. The entries in the vector G(S) = (G(x1,S),G(x2,S),...,G(x2,S)), where

i=1fi(t) = 1

xS

K(x)+S.

G(x,S) = c(x) · U(x,S)

define the per hour growth rates of each cell type at resource concentration S.

Throughout, cell densities computed from the mutation-selection chemostat equation (1) will be quoted in

units of log10(cells) per ml unless otherwise stated (and ‘log’ will denote the base-10 logarithm). By the term

quasispecies we mean a cluster of types whose densities form a unimodal distribution where a cluster is defined

by each type being a mutational neighbour of at least one other type in that cluster.

When we speak of the fitness landscape in the remainder, we refer to the two-dimensional surface defined

by the graph of the function G(x,S) for 0 ≤ x ≤ 1 and S ≥ 0, note that this depends both on cell type

through the normalised maximal uptake rate, x, but also on the resource concentration, S. There are natural

constraints on this function, denoted G for ‘growth’. For example, the availability of more resource should

lead to a higher fitness of each cell type and therefore

G(x,S1) > G(x,S2)whenever

S1> S2

should hold. In the complete absence of the limiting carbon source, S = 0, it is assumed that no cell can grow

and so G should satisfy

G(x,0) = 0for all

x ∈ [0,1].

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION5

These two properties will be assumed to hold in this article and, indeed, because of assumptions (A6) and

(A7) above we will use the following specific form of fitness landscape

(2)

G(x,S) = c(x) · Vmax

xS

K(x) + S;

we shall also impose the mild restriction that G is a smooth function of all its arguments. Throughout K or

K(x) will be called a half-saturation constant and 1/K or 1/K(x) will be called the affinity for resource and

K(x) will never be zero. Similarly, Vmaxwill always denote the maximum uptake rate of a carbon source into

a cell.

The definition in (2) can be used to encode two trade-offs within the MSC model. (i) If K(x) is an increasing

function of x, then the half-saturation constant of each cell type increases with increasing maximal uptake

rate, this is said to be a rate-affinity trade-off (see [27]). (ii) If c(x) is a decreasing function of x, then cell

yield decreases with increasing maximal uptake rate, this is said to be a rate-yield trade-off (see [32]). In this

framework, the fittest cells are those with the highest possible uptake rates of resource but they do pay a cost

for this in terms of having reduced yield, the cells with higher yields therefore have lower resource uptake

rates.

If we define the vector of type densities, b(t) = f(t)·∆(t) = (b1(t),b2(t),...,bn(t)), then the pair (b(t),S(t)),

with b(t) measured in units of cells per ml, satisfies the differential equation

d

dtb

d

dtS

=

?(M − I)b + (G(S) − d)b,

(3a)

=

d(S0− S) − ?U(S),b?

(3b)

and it is this form of the MSC equtation on which we concentrate throughout the remainder. Of particular

interest here is the question of how the diversity of solutions of (1) and (3) depends on system parameters, in

particular the mutation rate ?, with respect to some diversity measure H.

The mathematical theory of the chemostat described, for example, in [39, 8] does not apply directly to (3)

and although one can prove results on the global convergence to equilibrium for (3), one cannot easily exclude

the possibility that it supports periodic solutions and chaotic attractors. Such solutions are already known to

occur in simpler mutation-selection models than (1), for examples see [14]. As a result, estimates of long-term

diversity supported by equation (3) will be obtained by studying the steady-state of this equation and by a

posteriori numerical verification that the steady-states found in this way are locally and asymptotically stable.

A straightforward eigenvalue computation is sufficient to establish this in practise and these are performed

routinely in the computations that follow; while the algorithms developed in Matlab to rapidly solve (3) are

able to report on any absence of local stability, no violations of such conditions were encountered1. We could

go further and use techniques from rigorous computation to establish these statements rigorously, but this lies

outside the scope of the present manuscript.

3.1. The MSC model generates a dissipative dynamical system. Under the restrictions imposed on

equation (3), there is an efficiency constant C > 0 (independent of S,d,S0and ?) such that the inequality

G(S) ≤ C · U(S)

1In addition, changes in local stability induce visible changes in the geometry of equilibrium solution branches, namely

bifurcations, and it is shown below in Proposition 3 that no bifurcations are possible in non-trivial solution branches under certain

restrictions, a theoretical result entirely consistent with our numerical computations. Eigenvalues located on the imaginary axis

is a cause of oscillatory solutions and these were also tested for, but none were found.

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6R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

holds in a pointwise sense, a natural condition stating that the growth rate of any type is no larger than some

constant multiplied by uptake rate, in any environment. Define Σ = S0−S −1

time derivative of Σ:

?

= − d

C?1,b? and let us compute the

d

dtΣ = −d

dtS −1

?

C

1,d

dtb

?

= −d(S0− S) + ?U(S),b? −1

?

C?1,?(M − I)b + (G(S) − d)b?

S0− S −1

C?1,b?

+

?

U(S) −1

CG(S),b

?

= −dΣ + positive

≥ − dΣ.

It follows from this inequality that for any positive number η, there is a time T such that Σ(t) ≥ −η for all

t > T and so we obtain the following dissipative bound on the long-term behaviour of solutions of (3):

(4)

S(t) +1

C?1,b(t)? ≤ S0+ η

for all sufficiently large t. As a corollary of this argument, the inequality (4) holds for steady-state solutions

of (3) with η = 0.

3.2. The mutation matrix, M. We will assume throughout that the matrix M = (mij)n

matrix (meaning 1TM = 1T) with zero elements on the diagonal and non-negative elements on the off-

diagonal. The matrix M is constructed from the probabilities that a cell of type j exhibits a change of type,

to i, due to a mutational event. Unless explicitly stated otherwise, we will assume that M is irreducible,

meaning that for each pair (i,j) there is a power p = p(i,j) such that the (i,j)-th entry of Mpis non-zero.

This can be interpreted as a connectivity assumption on the set of types as it follows from this assumption

that there is a finite sequence of mutations that provides a path between any two types.

i,j=1is a stochastic

supplement fig. 2 – Differences between the parent and offspring maximal uptake rate

(and other cell phenotypes) arise when a mutation occurs, the parameter ? indicated on each

transition represents mutation rates, the indicated values 1 − ? and 1 − 2? are derived from the

probability that a change in maximal uptake rate does not occur. In (a) there is a small change

in uptake rate with each mutation, (b) is the same as (a) except that a mutational barrier is

introduced; in (a) and (b) the mutational neighbours of type i are types i−1 and i+1. If larger

changes in cell uptake rate occur with each mutation, this is represented by diagrams in (c)

and (d) and, note, the latter has a higher mutational connectivity than the former. In all cases,

the cell types coloured blue represent ‘type boundaries’ at the extreme low and high limits of

possible resource uptake rates.

low uptake typeshigh uptake types

(a)

(b)

barrier

1

2

34

5

(c)

(d)

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Remark 1. The so-called connectivity of mutations is determined from the pattern of non-zero entries in

the matrix M = (mij), type i is said to be connected to type j by a single mutation if mij is non-zero. For

example, the matrix denoted M above and associated with the mutational structures illustrated in supplement

fig. 2(a) and (b), respectively, are the following tridiagonal matrices:

10

2

00

0

2

0

2

0

00

2

01

000

2

0

Ma=

0

1

2

000

1

11

1

1

and

Mb=

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

2

0

0

1

0

1

2

;

note that the column sums of both matrices are unity. The former matrix is irreducible and this is the structure

of the mutation matrix used throughout this paper unless specified otherwise, suitably adapted to the appropriate

number of types, whereas the latter matrix, Mb, is not reducible by construction. To see this, note that the red

block of zeros in Mbarises because no mutation can occur between types ‘2’ and ‘3’ in supplement fig. 2(b).

For comparison, the mutational structures illustrated in supplement fig. 2(c) and (d) do not correspond to

tridiagonal matrices.

If the probability of mutation is the same for all cell types, ?, one can associate a Markov process M with

the transition rates between different cell types by defining

M =

no mutation

? ?? ?

(1 − ?)I +

mutation

????

?M

= I + ?(M − I).

If ? is a small quantity, this expresses the idea that most offspring cells express the same type as their parent,

as depicted in supplement figs. 2(a) and 2(b), but if a mutation does occur then the matrix M contains the

probabilities of a transition occurring to each of the other types. Thus M must have zeros on its main diagonal,

as do Maand Mbin the above examples.

One consequence of the irreducibility and non-negativity of M is the existence of a unique invariant density

ν = (ν1,...,νn)Tassociated with this mutation matrix, meaning that ν satisfies

(5)

Mν = ν

and

n

?

i=1

νi= 1,

where νj> 0 for all j.

In the case of a neutral fitness landscape whereby G(S) = g(S)1 for some scalar function g(·), but also

in the limit whereby the mutation rate ? is ‘infinite’, after re-scaling time if necessary and noting that f has

entries that sum to unity, equation (1a) reduces to the following linear diffusion equation

d

dtf = (M − I)f

whose solution operator e(M−I)tis mass-preserving. So, if f(0) has entries that sum to unity, it follows that

f(t) = e(M−I)tf(0) also has entries that sum to unity for all t ≥ 0 and so it converges to ν as t → ∞. As a

result, it is trivially possible to chose mutation matrices, M, in such a way that (1) has (in an approximate

sense) any unit vector f we desire as its equilibrium in a neutral landscape or when ? is large enough.

In order to avoid trivialities like this, whereby any equilibrium of (1) can be constructed, including very

diverse states, but only by virtue of a structure imposed within M, in the remainder we shall restrict attention

to those matrices M for which ν is equal to the uniform distribution1

n1. We shall also impose a non-neutrality

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8R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

condition on our fitness landscape: mathematically speaking, for each pair of distinct indexes, i and j, by this

we mean that Gi(S) ?= Gj(S) for infinitely many S.

We re-iterate an assumed property of all the numerical computations and simulations conducted in this

supplementary article: M will denote a process whereby a single mutational event forces a change of type

of one unit in type space. As a result, M will be a tridiagonal matrix and it will have

density. We will utilise a working assumption that M is reversible a.k.a symmetric, meaning M = MT

(equivalently mij= mji) to help simplify the theoretical presentation that follows. This assumption can be

substantially weakened in a number of directions that we discuss later. Do note that when one assumes M

to be an irreducible, non-negative, stochastic matrix, symmetric matrix, because MT1 = 1 and M = MTit

necessarily follows that ν must be equal to the uniform distribution, ν =1

1

n1 as its invariant

n1.

3.3. Comment: mutations are measured per unit time. The structure of the MSC equations in (1)

ensures that this model does not represent mutations originating from DNA replication errors alone as it

predicts changes in cell type even when cell division does not occur, namely when setting G(S) ≡ 0. The

resulting linear differential equation

types2. However, the phenotype of a cell, broadly defined, may indeed be dynamic in time and due, for

example, to regulatory changes or to mutations induced by UV radiation. Stresses on the cell, the introduction

of mutagens or the suppression, for example, of the SOS cycle may mean that biologically-realistic mutation

rates are themselves dynamic quantities and subject to selection, a property not contained within the MSC

model. Consistent with prior population genetics models (see for example [14]) the mutation rate ? in (1) is

measured per unit time (in fact, per individual cell per unit time) and it is a fixed, unchanging quantity for

all cell types.

As pointed out in [44, p. 1018] in the context of Ising models of sequence evolution, measuring mutation

rates either per cell division or per generation requires a different model structure than if they were measured

per unit time. As a result, we propose the following alternative equation to (3) that we do not analyse here:

let (b(t),S(t)) satisfy the differential equation

d

dtb = ?(M − I)b − db is a diffusion equation that can generate novel

d

dtb

d

dtS

=(I + ˜ ?(M − I))(G(S)b) − db,

(6a)

=

d(S0− S) − ?U(S),b?,

(6b)

where I is the identity matrix. If we were to set G(S) ≡ 0 in equation (6a), we would obtain the exponential

decay equation

setting the mutation rate ˜ ? equal to zero in (6) recovers the competitive chemostat equations, as it must.

The matrix I+˜ ?(M −I) in (6) can be written (1−˜ ?)I+˜ ?M which expresses the idea that the dimensionless

quantity ˜ ? is the probability of mutation and then M, whose columns sum to unity, identifies the change in cell

type that subsequently occurs from that mutation. Like (3), equation (6) generates a dissipative dynamical

system, an observation that may be exploited to produce conditions under which both classes of model have

globally attractive, non-trivial steady-state solutions.

d

dtb = −db which contains no diffusion terms and so generates no novel cell types, furthermore

2If the support of a vector b = (b1,...,bn) is the set of all indexes j ∈ {1,2,...,n} such that bj?= 0, the differential equation

dtb = F(t,b) is said to generate no novel types if the support of b(t) is equal to that of b(0) for all t ≥ 0. If there is a t such

that the support of b(t) and b(0) are different, then the equation is said to generate novel types.

d

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION9

4. Trade-offs: theory and evidence

4.1. Rate-yield trade-off: definitions. A substrate, such as glucose, is converted to ATP at a certain rate

per unit time, this in turn is converted to biomass and hence to cellular growth. In [2], the authors observe

that if microbes are limited by their energetic resource, the amount of biomass formed per unit of ATP is

approximately constant and does not depend on the mode of ATP production. Therefore, as highlighted in

[33], if the rate of ATP production increases, the rate of biomass formation and thus the growth rate of an

organism also increases. Thus the ‘rate’ component of a rate-yield trade-off can be variously measured as moles

of substrate per unit time, moles of ATP per unit time, biomass production per unit time or growth rate,

computed from cell divisions, per unit time. Yield, by contrast, considers production per unit of substrate.

This again can be moles of ATP per mole of substrate, biomass per mole of substrate or number of cell

divisions per mole of substrate.

supplement fig. 3 – Linear and sigmoidal rate-yield trade-offs for soil microbes at three

different temperatures: in all three cases we fit both a linear function and a nonlinear function

whereby the yield equals a − bR + c/(1 + exp(d + eR)), R denotes growth rate. In the middle

plot the 95% confidence interval for parameter c is (0.043,0.108) which does not contain zero,

thus indicating a better fit for the nonlinear form (data taken from [18]).

468 10

−3

x 10

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

growth rate (per hour)

yield (bacteria dry mass/substrate mass)

Growth rate−yield trade−off (0 − 4 o)

empirical data

nonlinear fit

linear fit

0.020.03

growth rate (per hour)

0.040.05 0.06 0.07

0.35

0.4

0.45

0.5

0.55

yield (bacteria dry mass/substrate mass)

Growth rate−yield trade−off (14 o)

empirical data

nonlinear fit

linear fit

0.040.05

growth rate (per hour)

0.06 0.07

0.4

0.42

0.44

0.46

0.48

0.5

0.52

yield (bacteria dry mass/substrate mass)

Growth rate−yield trade−off (22 o)

empirical data

nonlinear fit

linear fit

4.2. Rate-yield trade-off: evidence. Perhaps the best description of this trade-off and of its form comes

from yeast, see [45], supplement figure 5(a) and [19, Figure 1] where the latter provides a summary of rate-

yield trade-off data for various yeast species using data from [23, 35]. Evidence for the rate-yield trade-off

is provided by Novak et al. who examined E. coli and showed that when comparing individuals within a

population, such a trade-off was found [30]. Other studies indicate that maximal yield and maximal rate

are incompatible. Holophaga foetida, for example, can double its maximum specific growth rate at the cost

of a halved growth yield [17]. Likewise, Acetobacter methanolicus (see [28]) and Saccharomyces kluyveri (see

[25]) can increase growth rates by shifting catabolic flow into metabolic branches with lower biomass yield.

Pseudomonas fluorescens cultured in the chemostat has also been claimed to exhibit a rate-yield trade-off [12,

supplementary text].

While the form of the trade-off is well-described in yeast, the literature also describes this trade-off within a

single prokaryote species and within a community of microbes with sufficient detail to determine one or more

related nonlinear forms. As regards microbial communities, Lipson et al. [18] examine rate-yield relationships

for soil microbes under three temperature regimes and found a negative correlation between rate and yield in

each. For two cases a linear fit is best, however, at 14 degrees a function with a sigmoidal structure containing

both concave and convex regions provides a better fit to their data (see supplement fig. 3).

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10R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

supplement fig. 4 – (a) An empirically-determined rate-yield trade-off in E.coli derived from

data presented in [41]. A nonlinear Hill function is fitted to the data, illustrating a significant

sigmoidal form with concave and convex regions. (b) Empirical rate-yield data from [26] of

E.coli cultured on four different carbon sources also exhibits a significantly sigmoidal form. (If r

denotes the rate data and y the yield data, both scaled linearly by their minimum and maximum

values to lie between 0 and 1, the datafit in (a) uses the Hill function ‘y = a−br−cre/(de+re)’

whereas (b) uses the monotonic cubic model ‘y = a−br−c(r2/2+cr3/(12b))’ and the linear fit

is shown for comparison. Significantly, in all cases, the estimated 95% confidence interval for c

does not contain zero.)

(a)

1 1.52

0.44

0.45

0.46

0.47

0.48

0.49

0.5

growth rate (per hour)

yield (bacteria dry mass/substrate mass)

Rate−yield trade−off of E.coli

empirical data

nonlinear fit

linear fit

(b)

0.30.35 0.40.45 0.50.550.6

0.25

0.26

0.27

0.28

0.29

0.3

0.31

Rate−yield trade−off of E.coli on maltose

yield (bacteria dry mass/substrate mass)

growth rate (per hour)

0.40.50.6 0.70.80.9

0.235

0.24

0.245

0.25

0.255

0.26

0.265

on glucose

growth rate (per hour)

0.4 0.5

growth rate (per hour)

0.60.7 0.8

0.26

0.27

0.28

0.29

0.3

on mannitol

0.3 0.4 0.50.6 0.7

0.24

0.25

0.26

0.27

0.28

0.29

on sorbitol

growth rate (per hour)

empirical data

nonlinear fit

linear fit

Evidence for a negative relationship between growth rate and yield within one bacterial species, E. coli,

has existed in the classic microbiology literature for over half a century and by collating two sources we can

determine a possible form for this trade-off. Monod reported growth rate and cell yield of E. coli cultures

grown at a range of temperatures and a variety of substrates [26]. The individual relationships of rate and

yield versus temperature from this work are presented within a few pages of each other in the classic Bacterial

Metabolism by Stephenson [41], but to our knowledge these two experiments have not been integrated to

elucidate a possible relationship between rate and yield. It is known that the optimum growth temperature

of E. coli is close to 37oC. Noting that growth yield declines from suboptimal to optimal temperatures,

interpolating the data from the graphs of Figure 10, page 171 and Figure 16, page 178 in [41] to allow for a 1o

discrepancy in temperatures and merging the two data sets, this results in the negative and indeed sigmoidal

relationship shown in supplement fig. 4(a).

We can corroborate this with further evidence of a within-species rate-yield trade-off using data found in

Table 28, p. 110 and Table 30, p. 111 of [26] where the yield and rate experiments were performed on E. coli

at the same temperature and so no interpolation is necessary, moreover, the growth medium (milieu S) had

either glucose, sorbitol, mannitol or maltose as carbon source. We note that the fits to this data shown in

supplement fig. 4(b) all have a sigmoidal nonlinear form. We conclude from the evidence presented here that

in all the trade-off forms examined, including yeast, E. coli and soil communities, the rate-yield trade-off in

microbes may follow a sigmoidal structure and so possess at least one convex component.

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supplement fig. 5 – (a) An empirically-determined rate-yield trade-off in yeast indicates

that different uptake rates can correspond to equal, or nearly equal yields (curve derived using

data from [45]). (b) Empirical rate-affinity data in S. cerevisiae [9] is consistent with a sigmoidal

trade-off (uptake rate was log-transformed and the data fitted using ‘affinity = a − bR + c/(1 +

exp(d+eR))’ where R denotes the base-10 logarithm of uptake rate, the 95% confidence interval

for the parameter c is (0.0895,0.320) and the linear fit is shown for comparison). (c) This is

plot (b) reproduced with a linear scale for the uptake rate and where 1/K on the y-axis denotes

glucose affinity.

(a)

0510 15 20

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

uptake rate (mmol glucose /(g biomass × h))

yield (g biomass / mmol glucose)

Uptake rate−yield trade−off

empirical data

piecewise linear interpolant

(b)

0 0.20.4 0.60.81 1.2 1.4

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

log10 uptake rate (log(mmol glucose /(g biomass × h)))

glucose affinity (L/mmol)

Uptake rate−affinity trade−off

empirical data

linear fit

nonlinear fit

(c)

05 101520 25

0

0.05

0.1

0.15

0.2

0.25

uptake rate (mmol glucose /(g biomass × h))

1/K (L/mmol)

Uptake rate−affinity trade off

empirical data

nonlinear fit

supplement fig. 6 – (a) A cartoon form of the empirical rate-yield trade-off from supplement

fig. 5(a). (b) A possible trade-off obtained by extending (a) to a wider physiological regime.

(a)

growth rate

yield

(b)

growth rate

yield

Supplement fig. 5(a) displays an empirically-determined rate-yield trade-off obtained using data taken from

[45] illustrating that microbial cells with different uptake rates may exhibit very similar yields, supplement

fig. 15 below illustrates how this empirical curve is translated into a simplified form to be used within the

mathematical model (3). As a result, all of the trade-offs we use to understand the diversity properties

generated by solutions of equation (3) are shown in supplement figs. 3, 4, 5, 7, 8 and 15 where they exhibit

linear, sigmoidal and staircase forms. The staircase trade-offs encode some redundancy in the sense that the

same, or very similar, yield and affinity phenotypes may be expressed by cells with different uptake rates. Our

rationale for our occasional use of staircase or multiply sigmoidal trade offs is illustrated in supplement fig. 6.

We reason that trade-off data may take different forms when measured in different physiological regimes, as

can be seen in supplement fig. 3 and so, when extending the trade-off data given for example in supplement

fig. 5(a) and 5(b), we hypothesise that several regimes may exhibit sigmoidal behaviour. When this behaviour

is concatenated to form a single trade-off, it is possible that a staircase-like shape will result.

4.3. Rate-yield trade-off: biochemical basis. For several of the above cases, the biochemical basis of the

rate-yield trade-off involves alternative routings for metabolism that differ both in the ATP production and

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12 R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

supplement fig. 7 – The left-hand diagram in each plot is a rate-yield trade-off, the middle

plot is a rate-affinity trade-off and the right-hand diagram is the resulting fitness landscape:

(a) a convex rate-yield trade-off and (b) a sigmoidal rate-yield trade-off. (Darker regions in the

rightmost diagrams indicate lower resource concentrations and the highest peaks (approaching

the thick blue line) are only accessible when resources are abundant.)

(a)

0 0.20.4 0.60.81

5.8

5.9

6

6.1

6.2

6.3

6.4

log (yield)

Rate−yield trade−off used in the MSC model

normalised uptake rate

Vmax = 2e−06µ g / cell / h

0 0.2 0.40.60.81

0.092

0.094

0.096

0.098

0.1

0.102

0.104

0.106

0.108

0.11

Rate−affinity trade−off used in the MSC model

K

normalised uptake rate

0 0.20.4 0.60.81

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

fitness landscape

normalised maximal uptake rate

growth rate (h−1)

(b)

0 0.2 0.4 0.60.81

6

6.1

6.2

6.3

6.4

6.5

log (yield)

Rate−yield trade−off used in the MSC model

normalised uptake rate

Vmax = 2e−06µ g / cell / h

0 0.20.4 0.60.81

0.092

0.094

0.096

0.098

0.1

0.102

0.104

0.106

0.108

0.11

Rate−affinity trade−off used in the MSC model

K

normalised uptake rate

0 0.20.40.60.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

fitness landscape

normalised maximal uptake rate

growth rate (h−1)

the conditions under which they are employed. For example, E. coli can employ a less efficient fermentative

metabolism when glucose and oxygen are abundant. The Crabtree effect in yeast involving the diversion

of pyruvate to ethanol, rather than using the ATP-producing citric acid cycle, provides one example of

such an alternative routing. In this example, as with others, the biochemical pathways divide between a

long, ATP-high yield pathway and a short, low-yield overflow pathway [29], the high yield pathway being

utilised when resources are scarce. Similarly, the trade-offs in Holophaga foetida, Acetobacter methanolicus

and Saccharomyces kluyveri all involve shifts from catabolic into overflow metabolic branches with lower

biomass yield, when resources are abundant [28, 25].

In the absence of alternative routings, a rate-yield trade-off may yet be found. For example, with an abun-

dant substrate, anabolism and catabolism can be unbalanced. This can lead to ATP spillage [36] and may be

mediated by the usage of ATP in futile cycles, such as futile cycles of protons, again see [36]. More important,

possibly, are direct costs of toxic schemes of sugar usage. Most notable in this context is methylglyoxal, the

toxic product of intermediates of glycolysis through non-enzymic elimination of phosphate groups [36], this is

highly cytotoxic being both inhibitory of cell division and affecting potassium concentrations. It is thought

that if there were insufficient ADP to run normal glycolysis, for example when there is an excess of energy,

triose phosphates can be employed in the non-ATP-generating methylglyoxal shunt [5]. A consequence of this

is that when intermediates of glycolysis are abundant and cannot be processed sufficiently quickly, toxic prod-

ucts accumulate and possible ATP producing molecules not produced, thus reducing yield at high resource

concentrations. Others suggest that at high ATP production rates, rates of protein production must also be

high, which is likely to come at a direct cost owing to the reduced rate of production of other key proteins,

thus high rate can come at a cost in terms of yield [48].

4.4. Is the rate-yield trade-off likely to be common? While the above evidence suggests both that

the rate-yield trade-off can be found and what the underlying biochemical basis may be, it does not suggest

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION 13

supplement fig. 8 – (a) A nonlinear rate-yield and a linear rate-affinity trade-off, (b) the

same rate-yield trade-off as (a) and a different rate-affinity trade-off. The right-hand column il-

lustrates that the fitness landscape resulting from each trade-off combination depends less on the

form of the rate-affinity trade-off at higher resource concentrations, only at low concentrations

does the rate-affinity relationship matter.

(a)

0 0.2 0.40.60.81

5.9

6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

log (yield)

Rate−yield trade−off used in the MSC model

normalised uptake rate

Vmax = 1e−06µ g / cell / h

00.2 0.40.60.81

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Rate−affinity trade−off used in the MSC model

K

normalised uptake rate

0 0.20.40.6 0.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

fitness landscape

normalised maximal uptake rate

growth rate (h−1)

(b)

00.20.4 0.60.81

5.9

6

6.1

6.2

6.3

6.4

6.5

6.6

6.7

log (yield)

Rate−yield trade−off used in the MSC model

normalised uptake rate

Vmax = 1e−06µ g / cell / h

00.2 0.4 0.60.81

0

0.02

0.04

0.06

0.08

0.1

Rate−affinity trade−off used in the MSC model

K

normalised uptake rate

00.2 0.40.6 0.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

fitness landscape

normalised maximal uptake rate

growth rate (h−1)

whether such trade-offs are likely to be general, beyond the fact that the underlying metabolism, such as

overflow metabolism, is commonly observed. Two approaches have been taken to address this issue, both of

which suggest trade-offs are either inevitable or likely. Perhaps the most convincing approach is to argue that

tradeoffs are themselves evolutionarily optimal solutions. To see this, consider the problem of when low-yield

overflow metabolism is, and is not used [24].

There are several types of model that address this issue [24]. For example, if toxic metabolic products

build up under high resource conditions [20] it is simple to see why overflow metabolism, rapidly purging

the system, might be favoured. While this is logically robust, it is unclear how general an explanation this

might be. Molenaar et al. suggest what they propose to be a potentially more general explanation [24],

one that supposes there to be costs of metabolism, including the costs of manufacture of importer channels

and enzymes (see also [48]). Under low resource conditions, the costs of obtaining resources must be high

as resources are scarce, ensuring that the optimal strategy requires high ATP payback for a necessarily high

investment. Under resource abundance metabolites are cheap, it is thus not worth investing more to obtain

more ATP, rather the optimal strategy is high rate low yield, employing the overflow metabolism.

To illustrate this they consider a choice between a metabolically efficient (high ATP yield) pathway and a

catalytically efficient but metabolically inefficient pathway (low yield overflow). An optimal cell, they show,

shifts from the catalytically efficient to the metabolically efficient pathway when going from high to low

substrate concentrations. With decreasing substrate concentrations there comes a point at which channelling

substrate through a catalytically efficient resource-wasting pathway does not pay off anymore in terms of

growth rate, rather, high pay-off for the high investment is needed and hence routing through the slow, but

ATP rich, pathway.

This model correctly predicts not just that the ATP rich pathway should be used less under resource

abundance, but that it should be actively down-regulated to save costs [24]. Consistent with such models is

evidence that different optimality criteria apply at high and low resource levels, evidence that has come from

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14R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

fitting flux balance analysis models to empirical data at a range of resource levels [38]. As expected, given the

above, unlimited growth on glucose is best described by the maximisation of ATP yield per flux unit, while

under nutrient scarcity in continuous cultures, by contrast, the maximisation of the overall ATP or biomass

yields achieves the highest predictive accuracy.

An alternative line of argument postulates that rate-yield trade-off is a thermodynamic necessity [32] and

approaches including explicit kinetic descriptions of ATP production are argued to lead to similar results [40].

While data from Holophaga foetida is argued to be consistent with this model [17], the principle has been

asserted as operating only when the cell operates near thermodynamic equilibrium [34]. Nevertheless, if y

here denotes the yield of ATP per glucose, the function

?

is the flux obtained for a simple model of glycolysis [13, equation (43)] where k denotes Boltzmann’s constant,

T (here only) denotes temperature, ∆1and ∆2are certain free energy differences. As both ∆1and ∆2are

negative [13], this provides a kinetic derivation of a rate-yield trade-off, at least for the case of glycolysis.

JATP(y) = y

exp

?y∆1− ∆2

kT

?

− 1

?

4.5. Rate-affinity trade-off. Organisms may be subject to multiple trade-offs simultaneously and so, in

addition to the rate-yield trade-off described in some detail above, we will implement the so-called rate-

affinity trade-off in our models. Evidence for a trade-off between uptake rate and affinity has been observed

in S. cerevisiae [9] and E. coli [47] and data from the former has been used to produce supplement figs. 5(b

and c). This trade-off, whereby higher affinity of a receptor for a substrate comes at a cost of lower rate of

transportation of that substrate across the cell membrane is not peculiar to microbes and has already been

postulated as one possible reason why no single best strategy for resource competition has evolved in plants

[4, 11].

5. The bifurcation structure of steady-state solutions

Returning to the MSC equation into which we can encode the rate-yield and rate-affinity trade-offs discussed

in the previous section, we now fix the values of d and S0and consider ? to be a free parameter in equation

(3). Note that the steady-state problem of (3) can be written

0=

?(M − I)b + (G(S) − d)b,

d(S0− S) − ?U(S),b?,

(7a)

0= (7b)

and it will be helpful to cast equation (7a) as the S-dependent eigenproblem

(8)

d · b = (?(M − I) + G(S))b,

with d as the eigenvalue and b as the eigenvector. As M is assumed to be a symmetric, irreducible and

stochastic matrix, by the Perron-Frobenius theorem unity is its largest eigenvalue. Hence maxb?=0

and so

?(M − I)b,b?

?b?2

for every non-zero vector b that is not a scalar multiple of the invariant density of M.

The variational characterisation of eigenvalues and eigenvectors can now be used to show that the MSC

equation obeys the following optimsation principle in steady-state:

?Mb,b?

?b?2

= 1

= −1 +?Mb,b?

?b?2

< 0

(9)

d = max

b?=0

negative+

??

positive

??

???(M − I)b,b?

?b?2

+

?b,G(S)b?

?b?2

?

defines

=

ρ∗(?,S).

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION15

The eigenvector providing the relative frequency of different bacterial types that achieves the maximum in (9)

will be written throughout the remainder as b∗(?,S) and we shall assume for definiteness that it is normalised

so that ?1,b∗(?,S)? = 1. Note, when d and ? are fixed, equation (9), that is

(10)

d = ρ∗(?,S)

is a single equation for the equilibrium resource concentration S. To help make the notation a little more

transparent, notice that the two components of the steady-state (b,S) of (3) can be written in terms of one

another as b = λ · b∗(?,S), where λ is a proportionality constant whose significance will become apparent in

the following analysis.

The purpose of this analysis is to identify just two different bifurcation structures of (7) according to the

value of the parameter

dcrit(S0)

defines

=

1

n

n

?

i=1

Gi(S0).

Note that this function of S0, dcrit, is the mean absolute fitness of the population at the resource supply

concentration and this parameter plays a fundamental role in understanding the two bifurcation structures

alluded to above. The first of these structures can be stated as follows.

Proposition 1. Suppose that S0 > 0 is fixed and d < dcrit(S0), then for each ? > 0 there is exactly one

strictly positive steady-state (b(?),S(?)) of (3). Morever, as ? → ∞ the distribution of types converges to the

maximal diversity state (the uniform distribution) in the sense that

b(?)

?b(?),1?=1

As ? → 0 the distribution of types converges to the minimal diversity, competitive exclusion state in the sense

that if lim?→0b(?) ?= 0, then

lim

?→∞

for some permutation matrix P.

lim

?→∞

n1.

b(?)

?b(?),1?= P · (1,0,...,0)T

Proof. First fix S0> 0. The first part follows from the fact that the eigenproblem (8) can be re-written

(11)(d + ?) · b = ?Mb + G(S)b.

As M is irreducible and non-negative, for ? > 0 so too is the matrix ?M + diag(G(S)) and therefore, for

each S ≥ 0, the latter has an algebraically simple eigenvalue e(?,S) corresponding to a positive eigenvector

αb, where α > 0 is any constant. The matrix ?M + diag(G(S)) has entries that increase monotonically with

respect to both ? and S and so its dominant eigenvalue also increases monotonically with respect to both

variables. As a result, for each fixed d and ?, the resulting equation

d + ? = e(?,S)

has at most one solution for the steady-state sugar concentration S and equation (7b) then uniquely determines

the value of α. This argument ensures that when a solution of (7) exists with b non-negative, this solution is

unique.

We now establish the existence of steady-state solutions by solving (7). To achieve this, first set δ = 1/?

and then write b = θ1+v where v ∈ span{1}⊥in (7), so that ?v,1? = 0. As a result, we find that (7) defines

an equation F(v,θ,S,δ) = 0 where the mapping F : span{1}⊥× R3→ Rn+1is differentiable. Let us write

Π : Rn→ span{1}⊥for the orthogonal projection that satisfies Π[b] = v and Π[1] = 0. (Note that δ = 0

corresponds to ‘? = ∞’ in this formulation.)

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16R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

We now tackle the equation F(v,θ,S,δ) = 0, which can be written in full as

0=(M − I)v + δ · Π[(G(S) − d)(θ1 + v)],

?1,(G(S) − d)(θ1 + v)?,

d(S0− S) − ?U(S),θ1 + v?,

(∈ span{1}⊥) (12a)

0= (12b)

0=(12c)

and we do this by first setting δ = 0. This yields a solution of F = 0, namely F(v∗,θ∗,S∗,0) = 0 where

v∗= 0 ∈ span{1}⊥, θ∗=?U(S∗),1?

The existence of an S∗satisfying the latter in (13) necessitates a solution S∗≤ S0of the equation d = dcrit(S∗),

a solution that cannot exist if d > dcrit(S0). So, assuming d < dcrit(S0), the monotonicity of the function

dcrit(·) with respect to its argument and the fact that dcrit(0) = 0 yields the existence of an S∗< S0satisfying

d = dcrit(S∗), as required.

A short computation shows these conditions are sufficient for the derivative d(v,θ,S)F(v∗,θ∗,S∗,0) to be

an isomorphism from its domain of definition to its range and therefore the implicit function theorem can

be applied to deduce the local existence of a unique, smooth curve of solutions (v(δ),θ(δ),S(δ)) such that

F(v(δ),θ(δ),S(δ),δ) = 0 for all δ near zero. Now, the first component of the solution of (7), b, can be written

b(?) = θ(1/?)1 + v(1/?) where θ(0) = θ∗and v(0) = 0, so that b(?) = θ∗1 + O(1/?). This establishes the last

part of the proposition because b(?) is proportional to the vector 1 in the limit ? → ∞.

There remains to establish the existence of a solution of (7) for all ? > 0, equivalently for all δ > 0.

For this we can appeal to either modern global bifurcation theory [3] or to more classical Leray-Schauder

continuation results described in [49], either way we can deduce the existence of a continuum C (a maximal,

closed, connected set) of solutions of the equation F(v,θ,S,δ) = 0, unbounded in the solution measure

ν(v,θ,S,δ) = ?v?+|θ|+|S|+|δ|. So, each element of C has the form (v,θ,S,δ) and satisfies F(v,θ,S,δ) = 0.

Note that a maximum principle can be established using (11): if b ≥ 0 then from the irreducibility of M

we deduce that b ? 0, moreover S = 0 is not possible at any solution in C. From this it follows that if

(v,θ,S,δ) ∈ C and the vector b = v + θ1 contains any negative entries, by the connectedness of C there must

exist a non-negative (v?,θ?,S?,δ?) ∈ C for which b?= v?+θ?1 contains a zero entry. However, we then deduce

that b?= v?+ θ?1 ≥ 0 has a zero entry, but this is a contradiction as b?must satisfy (11) with S = S?≥ 0.

As a result, for any (v,θ,S,δ) ∈ C there results v + θ1 ? 0,S > 0. In other words, C defines an unbounded

continuum of strictly positive solutions of (7).

From (4), ?v?+|θ|+|S| must be bounded within C and so this continuum must be unbounded with respect

to the solution measure ν(v,θ,S,δ) = |δ| = |1/?|. As a result, the set C contains strictly positive solutions of

(7) associated with arbitrarily small values of ?, that is, with arbitrarily large values of δ. As C is connected,

with solutions of arbitrarily large and small, positive ?, the set C contains a solution of (7) for every value of

? > 0.

Finally, as ? → 0 along any solution sequence within C, the component b(?) must converge to a solution,

b, of the equation (G(S) − d)b = 0.

G(S)−dI can only have one zero entry on the main diagonal and so b must have the minimal-diversity form

(0,0,...,0,∗,0,...,0) where ∗ is a non-negative number. The result now follows.

The second bifurcation structure is described in the following result.

(13)

d(S0− S∗)> 0where

1

n?G(S∗),1? = d.

However, because G(·) is strictly monotone, the diagonal matrix

?

Proposition 2. Suppose that S0> 0 is fixed and d > dcrit(S0). There is a differentiable function ?crit(d,S0)

such that if ? ∈ (0,?crit(d,S0)), then there is exactly one strictly positive steady-state (b(?),S(?)) of (3).

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION 17

However, if ? > ?crit(d,S0) then the trivial steady-state (b,S) = (0,S0) is globally stable and eventually

exponentially stable for the dynamics of (3). As a result, there exists a region in the (d,?) plane whose

boundary is given by the graph of a function d = J(?) and the line d = dcrit(S0) in which the cell population

cannot persist (see supplement fig. 9).

Proof. For the same reason as explained in the proof of Proposition 1, when (3) has a non-zero steady-state

solution, it is unique. Now, fix S0> 0 and write the steady-state equation (7) in the form F(b,S,?) = 0 by

suitably defining a smooth mapping F and note the existence of a trivial solution branch of this equation:

F(0,S0,?) = 0 for all ? ≥ 0.

Consider the linearisation of F about this trivial solution with respect to (b,S), evaluated at (0,S0):

d(b,S)F(0,S0) =

?(M − I) + diag(G(S0)) − dI

0

−U(S0)T

−d

.

Applying the theorem on bifurcation from a simple eigenvalue [3], the equation F(b,S,?) = 0 undergoes a

biologically-relevant bifurcation from the trivial solution when there is a non-zero vector k ≥ 0 such that

[?(M − I) + diag(G(S0)) − dI]k = 0, that is

(14)

?(M − I)k + G(S0)k = dk

Using the variational characterisation of the eigenvalues of the matrix in (14), we must find an ? such that

?

Now, the function J(·) so-defined satisfies

J(0) = maxand

where

?k,1? = 1.

d = max

k?=0

??k,(M − I)k?

?k,k?

+?k,G(S0)k?

?k,k?

?

defines

=

J(?).

1≤i≤nGi(S0)

J(0) > J(?) ≥?1,G(S0)1?

?1,1?

= dcrit(S0)

for all ? > 0 as J(·) is a monotone decreasing function.

It follows that either (i) there is a unique solution of J(?) = d, or else (ii) there is no solution at all of this

equation because d < dcrit(S0). Case (ii) arises in the situation described in Proposition 1 in which case there

is no bifurcation of solutions from the trivial solution branch and an unbounded set of non-trivial solutions

can be globally defined (the set C defined in the proof of Proposition 1). We are therefore left with (i), in

which case to each d we can associate a value of ? such that J(?) = d, let us call this solution ?crit. We

are now able to define the function ?crit = ?crit(S0,d) given in the statement of the proposition using the

inverse function theorem and it satisfies J(?crit(S0,d)) ≡ d. In this case, the theorem on bifurcation from a

simple eigenvalue [3] can be applied to demonstrate the existence of a locally unique solution branch B of the

equation F(b,S,?) = 0 that extends from the trivial solution at the bifurcation point ? = ?crit(S0,d).

In order to describe the region in the (d,?)-plane at which the bifurcation of B occurs from the trivial solution

branch, namely the 1-dimensional set whereby J(?) = d, consider the following. By the Perron-Frobenius

theorem, note that the largest real eigenvalue d+? of the non-negative, irreducible matrix ?M +diag(G(S0))

corresponding to the eigenvector k ≥ 0 in equation (14) is algebraically simple when J(?) = d. Therefore, we

can apply the implicit function theorem to deduce that because J(?crit(S0,d)) = d identically, the function

?crit(·,·) is smooth in its arguments.

Let δ = 1/? and define τ(δ) = δ · J(?). Noting that k defined in (14) also depends smoothly on ? (see [3]

for a derivation of the fact that algebraically simple eigenvalues and eigenvectors depend smoothly on system

parameters) we will write k(δ) to emphasise this smooth dependence. Now differentiate the expression

?(M − I)k + G(S0)k = J(?)k

i.e.(M − I)k + δG(S0)k = τ(δ)k

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18R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

with respect to δ. Using a prime (?) to denote the derivative with respect to δ, this gives

(15)(M − I)k?+ G(S0)k + δG(S0)k?= τ?(δ)k + τ(δ)k?.

When taking the limit δ → 0, because (M − I)1 = 0 to remove any singularities we must set k(0) =1

τ(0) = 0, now taking the inner product of (15) with the vector 1 yields τ?(0) =?1,G(S0)

and so

τ(δ)

δ

δ→0

This aspect of the proof is illustrated in supplement fig. 9 where the region in the (d,?) plane for which

J(?) = d is illustrated as a thick black line.

That the non-trivial, bifurcating solution branch B contains the unique, non-trivial solution of (7) only for

0 < ? < ?crit(S0,d) is proven as follows. Recall that S0> 0 and d are given with d > dcrit(S0). Now suppose

that ? > ?crit(S0,d) and let κ > 0 be the vector that satisfies ?(M −I)κ+(G(S0)−J(?))κ = 0 and note that

d = J(?crit(S0,d)) > J(?) because ? > ?crit(S0,d) and J is a decreasing function. Let us compute

along solutions of (3):

?d

= ??(M − I)b(t) + (G(S) − d)b(t),κ? = ??(M − I)κ + (G(S) − d)κ,b(t)?,

= ??(M − I)κ + (G(S) − G(S0) + G(S0) − J(?) + J(?) − d)κ,b(t)?,

= ?(G(S) − G(S0) + J(?) − d)κ,b(t)?,

= ?(G(S) − G(S0))κ,b(t)? + (J(?) − d)?κ,b(t)?.

n1 and

?/?1,1? = dcrit(S0),

lim

?→∞J(?) = lim

δ→0

= lim

τ(δ) − τ(0)

δ − 0

= τ?(0) = dcrit(S0).

d

dt?b(t),κ?

d

dt?b(t),κ? =

dtb(t),κ

?

= ??(M − I)b(t) + (G(S) − d)b(t),κ?,

From (4), for all η?> 0 we can find a T such that for all t > T, S(t) ≤ S0+ η?. By the monotonicity and

continuity of G(·), for all η > 0 we can find a T such that for all t > T, G(S(t)) ≤ G(S0) + η and we choose

the positive quantity η =1

2(d − J(?)). It follows that

dt?b(t),κ? = ?(G(S) − G(S0))κ,b(t)? + (J(?) − d)?κ,b(t)? ≤1

d

2(J(?) − d)?κ,b(t)?

and so there are constants a,b > 0 that depend on T such that for all large enough t, ?b(t),κ? ≤ a·e−b·twhence

limt→∞b(t) = 0 eventually exponentially because κ ? 0. This establishes the last part of the proposition and

also shows that non-negative solutions of (7) only exist if ? < ?crit(S0,d) whenever d > dcrit(S0). Standard

global bifurcation arguments can now be used to show that B can be extended to all values of ? for which

0 ≤ ? < ?crit.

?

Remark 2. All numerical computations performed beyond this point were obtained using a pseudo arc-length

strategy [16, 6] to rapidly determine equilibrium states of the MSC equation with ? as the bifurcation parameter.

This numerical technique is an extension of Newton’s method that can cope with the presence of multiple

solutions, bifurcations, phase transitions and rapid continuous transitions between different solutions.

5.1. Lethal Mutagenesis. Proposition 2 identifies a relationship between S0 and d whereby lethal muta-

genesis may be a successful antibacterial strategy in (3). Namely, if d >

by raising the mutation rate ? it is possible to push the population towards, and beyond a bifurcation point

whereafter the fate of each cell is to be lost from the chemostat because the cells cannot divide sufficiently

quickly. The red area in supplement fig. 9(b) illustrates a parameter region where lethal mutagenesis can be

1

n

?n

i=1Gi(S0) = dcrit(S0), then

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION19

supplement fig. 9 – (a) The key ingredients in the proof of Proposition 2: it highlights a

region in the (d,?) plane where the evolving population can be driven to extinction by increasing

mutation rates. (b) An instance of (a) computed numerically using the MSC equation.

(a)

extinction

lethal mutagenesis

fails

0

lethal

mutagenesis

succeeds

graph of

(b)

0.81 1.2 1.4 1.6 1.8

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

dcrit(S0)

d

ε

Successful lethal mutagensis if d > Σj=1

n Gj(S0)/n ≈ 1.47 h−1

lethal mutagenesis fails extinction

successfully applied and supplement fig. 10(b)) illustrates the reduction in bacterial density obtained from

increasing the mutation rate.

Supplement fig. 10(b) also shows that as the mutation rate is increased, a bifurcation point is eventually

encountered at ? = ?critbeyond which extinction is assured. This bifurcation point is readily shown to be

a transcritical bifurcation in the sense of [1]. This establishes the result that along the equilibrium solution

branch that supports non-zero cell densities illustrated in the left-hand plot of supplement fig. 10(b), there

is a region within which bacterial density, ?1,b(?)? =?n

bifurcation point as ? increases. (This is particularly clear in supplement fig. 10(b) at the label marked TB.)

Standard bifurcation theory gives the existence of an estimate of the form

i=1bi(?), reduces approximately linearly near the

(16)

n

?

i=1

bi(?) = Constant · |? − ?crit| + O?(? − ?crit)2?

as ? → ?critfrom below but the region over which this approximation is valid, near to where log(?) ≈ −2, is

very narrow in supplement fig. 10(b). As the mutation rate increases, the cell population is almost constant

in density up until the point where a rapid and critical decline finally occurs. This rapid decline may well

be system-specific, but the linear approximation (16) is a universal feature of equation (3) within (A1-A7)

and the set of assumptions given in Proposition 2; such a linear reduction in density has been found in other

theories of successful lethal mutagenesis [22].

Conversely, Proposition 1 states that lethal mutagenesis may fail as an antibacterial strategy if d < dcrit(S0).

In this case, raising mutation rates may still reduce the density of cells in the chemostat but there is no

bifurcation point at which extinction is assured and the population will be driven towards a stable, diverse

state. The grey regions in supplement fig. 9 illustrate parameter regions where lethal mutagenesis may fail

and supplement fig. 10(a) provides a numerical example of this: note how cell densities do not fall below

approximately 106.44cells per ml in this computation.

6. Predictions of the MSC model: the coexistence hypothesis

We are now interested in mutation rates in the MSC equation that are not so high that the population

is driven towards extinction, as described in Proposition 2, nor towards the high-diversity state described in

Proposition 1. Moreover, Proposition 1 informs us that very low mutation rates are associated with a state of

competitive exclusion where survival of the fittest applies. As we seek to understand how the MSC model can

support the coexistence of different quasispecies of cell types, we must turn our attention to a cell population

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20 R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

supplement fig. 10 – Respectively, (a) and (b) are numerical illustrations of the results

contained in Propositions 1 and 2 when rate-yield and rate-affinity trade-offs are invoked si-

multaneously. The relative frequencies of cell types in the population are shown as red and

blue bar diagrams. The antibacterial strategy of lethal mutagenesis is successful in diagram (b)

as equation (7) possesses a transcritical bifurcation at a critical mutation rate ?crit (shown as

the label TB where log10(?crit) ≈ −2) whereas (a) does not possess such a bifurcation and the

strategy fails. Common with all transcritical bifurcations, when it occurs the solution branch

emanates from the bifurcation point at TB as a locally linear path.

(a)

−5−4 −3−2−10

6.4

6.42

6.44

6.46

6.48

6.5

6.52

6.54

6.56

6.58

log10(ε)

log10 density

Steady−state densities

00.2 0.40.6 0.81

0

0.5

1

S/S0 = 0.014901 , mutation rate = 0.3716

normalised uptake rate

relative frequency

00.2 0.4 0.60.81

0

0.5

1

S/S0 = 0.0072505 , mutation rate = 0.0034342

normalised uptake rate

relative frequency

(b)

−5−4−3−2 −1

−1

0

1

2

3

4

5

6

log10(ε)

log10 density

Steady−state densities

TB

00.20.40.60.81

0

0.5

1

S/S0 = 0.9966 , mutation rate = 0.0098512

normalised uptake rate

S/S0 = 0.39175 , mutation rate = 0.00050272

relative frequency

0 0.20.40.60.81

0

0.5

1

normalised uptake rate

relative frequency

at ‘intermediate’ mutation rates where mathematical and analytic results are hardest to obtain. As a result,

we now resort to numerical computations with the one goal in mind of finding conditions under which the

‘fittest’ and the ‘flattest’ can coexist in the system of equilibrium equations (7).

6.1. Steady-state sugar consumption decreases with increasing mutation rate. Equation (10) can

be used to probe the relation between the mutation rate and consumption of the limiting carbon source

in steady-state. Assuming (b(?),S(?)) to be an equilibrium of (3) parameterised by mutation-rate, upon

differentiating (10), that is ρ∗(?,S(?)) ≡ d, with respect to ? we find that

dS

d?(?) = −?b(?),(M − I)b(?)?

?b(?),G?(S(?))b(?)?> 0.

From this inequality we deduce that the equilibrium value of consumed resource in the chemostat reduces as

mutation rate increases.

Although a decline in consumed resource is expected to be correlated with a reduction in population density,

thus providing a rationale for the use of lethal mutagenesis, we saw in the previous section that this logic may

fail. Whether or not the population density declines depends crucially on the nature of the specific trade-offs

that form the fitness landscape. Inspect, for example, supplement fig. 11 wherein population densities can

increase with an increasing mutation rate (the fitness landscape shown in supplement fig. 8(a) is derived using

both rate-yield and rate-affinity trade-offs and this landscape was used for the computation that produced

supplement fig. 11).

6.2. Cell density scales linearly with S0. Equation (10) can be viewed as an equation for S that can be

solved (via the implicit function theorem) for S as a function of (d,?) and, as a result, it is clear that the

equilibrium value of S is independent of S0in the MSC model. Put more simply, once d and ? are fixed, so

too is S whatever value S0may happen to be.

This is a feature the MSC model (3) has in common with the chemostat models presented in [39] and it

can be exploited to make a prediction: using (7), as S depends only on (d,?), so too does the eigenvector b in

(7a) up to the scalar multiple λ defined earlier in the text that we can now determine from (7b). Indeed, b

must be proportional to the vector b∗(?,S(d,?)) as defined in §5 and it follows that we can write the vector b

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION21

supplement fig. 11 – Although an increase in mutation rates decreases resource consumption

at equilibrium in the MSC model, trade-offs can conspire to create increases in density by locally

flattening fitness peaks. Hence, as this example shows, population density may increase when

mutation rates increase.

−6.2 −6.1−6−5.9−5.8

6.37

6.38

6.39

6.4

6.41

6.42

6.43

6.44

log10 density

log10(ε)

Steady−state density

−6.2−6.1 −6−5.9−5.8

0

0.2

0.4

0.6

log10(ε)

Simpson’s index (normalised)

0 0.20.4 0.60.81

0

0.5

1

S/S0 = 0.17527 , mutation rate = 1.3485e−06

normalised uptake rate

S/S0 = 0.16102 , mutation rate = 9.6248e−07

relative frequency

00.20.4 0.60.81

0

0.5

1

normalised uptake rate

S/S0 = 0.14687 , mutation rate = 7.6713e−07

relative frequency

00.20.4 0.60.81

0

0.5

1

normalised uptake rate

relative frequency

in (7a) as b = λ · b∗(?,S(d,?)), including the constant λ. In order to determine the value of λ, using (7b) we

find that

λ = λ(S0,d,?) =

d(S0− S(d,?))

?U(S(d,?)),b∗(?,S(d,?))?.

Thus, provided there is sufficient resource to support the population, and S0> S(d,?) is a critical condition

needed for this, biomass scales approximately linearly with the resource supply concentration in the sense

that

?b,1? = λ(S0,d,?) = Constant · S0+ O(1)

as S0→ ∞; here O(1) is standard notation for any bounded function of S0.

This analysis highlights a non-physical feature of chemostat models like (1), there is no explicit ‘volume’

parameter of the culture vessel and we simply measure cells per unit volume. However, volume effects are

real and it cannot be true that biomass scales linearly with resource supply for arbitrarily large values of

S0, eventually space itself will be limiting. Spatial effects are observed in chemostat experiments and poly-

morphism can be maintained between producers and non-producers of adhesins that allows cells to stick to

the vessel wall or to other cells (see, for example, [50]). As bacterial populations can maintain a diversity of

types due to spatial structure, it is important that we do not introduce spatial effects into (1) but with this

stipulation there must come the caveat that our models cannot describe experimental devices at high sugar

concentrations. Nevertheless, the prediction that biomass scales linearly with S0stated above is borne out in

practise (see, for example, [15, Fig. 1] for E.coli limited by glucose).

6.3. Diversity is independent of S0. With the discussion of the previous section in mind, we now suppose

that H is a diversity measure on n-types, such as the Simpson index HS. We imposed the condition that any

diversity measure should be scale-invariant in §1 and so the diversity of the population in the chemostat at

equilibrium given by

H(b(S0,d,?)) = H(λ(S0,d,?) · b∗(?,S(d,?))) = H(b∗(?,S(d,?))),

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22R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

a number that is clearly independent of the supply concentration S0. Our observation here is consistent with

a prediction from Thompson’s Geographic Mosaic Theory [42] which states that diversity is constant across

environmental gradients for systems having only G×E interactions and, indeed, the MSC equation is a model

with this type of interaction.

supplement fig. 12 – Bar diagrams of the relative frequencies of cell types illustrating co-

maintenance of the fit and the flat at intermediate mutation rates in the MSC model when there

is (a) no rate-affinity trade-off but a convex rate-yield trade off (shown in supplement fig. 7(a)),

(b) a linear rate-affinity trade-off with a convex rate-yield trade-off (resulting fitness landscape

not shown). (b/right column/middle) A superposition of Gaussians has been plotted as black

lines to highlight each quasispecies.

(a)

−2.75 −2.7 −2.65

log10(ε)

−2.6−2.55−2.5

4.1

4.15

4.2

4.25

4.3

4.35

4.4

4.45

4.5

4.55

4.6

log10 density

Steady−state density

−2.75−2.7−2.65

log10(ε)

−2.6 −2.55−2.5

0

0.5

1

multimodality index

0 0.20.4 0.60.81

0

0.5

1

S/S0 = 0.79261 , mutation rate = 0.0024884

normalised uptake rate

S/S0 = 0.78398 , mutation rate = 0.0021785

relative frequency

00.20.4 0.60.81

0

0.5

1

normalised uptake rate

S/S0 = 0.77645 , mutation rate = 0.002113

relative frequency

00.2 0.40.6 0.81

0

0.5

1

normalised uptake rate

relative frequency

(b)

−2.75−2.7−2.65−2.6−2.55

5.8

5.85

5.9

5.95

6

6.05

6.1

6.15

log10 density

Steady−state densities

−2.75−2.7−2.65 −2.6−2.55

0

0.5

1

log10(ε)

multimodality index

0 0.2 0.40.60.8 1

0

0.5

1

S/S0 = 0.08614 , mutation rate = 0.002551

normalised uptake rate

S/S0 = 0.084045 , mutation rate = 0.0019424

relative frequency

0 0.20.40.60.81

0

0.5

1

normalised uptake rate

S/S0 = 0.081516 , mutation rate = 0.0018272

relative frequency

00.20.4 0.60.81

0

0.5

1

normalised uptake rate

relative frequency

supplement fig. 13 – Co-maintenance of the fit and the flat at intermediate mutation rates

in the MSC model. (a) No rate-affinity trade-off and a sigmoidal rate-yield trade off was used

(see supplement fig. 7(b)) and two datasets are shown: when n = 14 cell relative type frequencies

are shown as histograms, when n = 100 relative frequencies shown as lines in the right-hand

column. (b) The trade-offs illustrated in supplement fig. 8(a) were used to produce this diagram

showing it has a wider coexistence zone than in (a). (b/right column/middle) A superposition

of Gaussians has been plotted as black lines to highlight each quasispecies.

(a)

−3.1−3−2.9−2.8

log10(ε)

−2.7−2.6 −2.5

4.75

4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

5.2

5.25

log10 density

Steady−state density

−3.1 −3−2.9−2.8

log10(ε)

−2.7−2.6−2.5

0

0.5

1

multimodality index

00.20.4 0.6 0.81

0

0.5

1

S/S0 = 0.31155 , mutation rate = 0.0024289

normalised uptake rate

S/S0 = 0.30305 , mutation rate = 0.0013946

relative frequency

0 0.20.40.60.81

0

0.5

1

normalised uptake rate

S/S0 = 0.30023 , mutation rate = 0.0012524

relative frequency

0 0.20.40.60.81

0

0.5

1

normalised uptake rate

relative frequency

(b)

−3.8−3.6 −3.4−3.2 −3

6.26

6.265

6.27

6.275

6.28

6.285

6.29

6.295

6.3

log10 density

Steady−state densities

−3.8−3.6−3.4

log10(ε)

−3.2 −3

0

0.5

1

multimodality index

0 0.20.4 0.60.81

0

0.5

1

S/S0 = 0.28057 , mutation rate = 0.00071098

normalised uptake rate

S/S0 = 0.26276 , mutation rate = 0.00035453

relative frequency

00.20.4 0.60.81

0

0.5

1

normalised uptake rate

S/S0 = 0.24666 , mutation rate = 0.00019964

relative frequency

00.20.4 0.60.81

0

0.5

1

normalised uptake rate

relative frequency

6.4. The MSC model can maintain both fit and flat quasispecies. A fourth prediction of the MSC

model under our assumptions is the coexistence hypothesis: if survival of the fittest prevails at low mutation

rates and survival of the flattest is found at high mutation rates, under the structural impositions placed upon

equation (3) there exists at least one interval of intermediate mutation rates for which both the fit and the

flat coexist in steady-state.

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION23

If fit and flat quasispecies are dominant at low and high mutation rates respectively, the co-maintenance

principle to which we allude is a consequence of the continuous dependence of the unique steady-state of (3) on

the mutation rate. Namely, as we change that mutation rate, we continuously morph the density of types from

a unimodal fit distribution into a unimodal flat distribution, at some interval of intermediate mutation rates

we expect to observe a bimodal distribution of both fit and flat types. If the existence-uniqueness property of

this steady-state were to fail, as illustrated in supplement fig. 16 below, then the coexistence hypothesis can

fail.

supplement fig. 14 – (a) A resource-dependent fitness landscape encoding both rate-yield

and rate-affinity trade-offs (for the trade-offs see supplement fig. 8(b)) where darker regions

correspond to lower resource concentrations. (b) Branching of an initially monomorphic qua-

sispecies (the red dot) is traced through this fitness landscape: the red line denotes the equi-

librium fitness landscape upon which two quasispecies are stably maintained. (c) The passage

from fittest to flattest with increasing mutation rate for the fitness landscape depicted in (a)

passes through an interval of coexisting quasispecies. Here, a monotone decrease in density with

increasing mutation rates is observed.

(a)

00.20.40.60.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

fitness landscape

normalised maximal uptake rate

growth rate (h−1)

(b)

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2

max. uptake rate (normalised)

fitness landscape

sugar concentration

growth rate

(c)

−5−4.9 −4.8−4.7−4.6−4.5 −4.4

6.2

6.25

6.3

6.35

6.4

6.45

6.5

6.55

log10 density

Steady−state density

−5−4.9−4.8−4.7

log10(ε)

−4.6−4.5−4.4

0

0.2

0.4

0.6

0.8

1

multimodality index

00.20.40.60.81

0

0.5

1

S/S0 = 0.28706 , mutation rate = 4.0973e−05

rel. freq.

0 0.20.40.60.81

0

0.5

1

S/S0 = 0.22543 , mutation rate = 2.3291e−05

rel. freq.

00.20.40.60.81

0

0.5

1

S/S0 = 0.1913 , mutation rate = 1.4329e−05

rel. freq.

0 0.2 0.40.60.81

0

0.5

1

S/S0 = 0.17512 , mutation rate = 1.2056e−05

rel. freq.

00.20.40.60.81

0

0.5

1

S/S0 = 0.14989 , mutation rate = 9.7302e−06

normalised uptake rate

rel. freq.

The co-maintenance of two quasispecies is illustrated in supplement figs. 12 and 13 whereby the mutation

rate is increased and three different type distributions are illustrated, at low, intermediate and high mutation

rates. Supplement fig. 12(a) was produced when the only trade-off implemented in the MSC model was the

rate-yield trade off, as a result all the cell affinities for the carbon source are equal but the rate-yield trade-off

follows a convex form. To produce supplement fig. 12(b), a linear rate-affinity trade-off was invoked in addition

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24R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

to the convex rate-yield trade-off, this has the effect of widening the interval of mutation rates for which we

obtain coexistence.

Supplement fig. 13(a) was obtained using a sigmoidal form for the rate-yield trade-off, a form found for

the natural soil microbes illustrated in supplement figs. 3 and 4, but no rate-affinity trade-off was used

here. Supplement fig. 13(b) was obtained using the theoretical landscape shown in supplement fig. 8(a)

and supplement fig. 13(b,right column,middle plot) illustrates how two approximately normally distributed

quasispecies can persist across a range of mutation rates.

Supplement figs. 12, 13 and 14 use a multimodality index (the ratio of biomass contained within each

quasispecies) to illustrate a window of mutation rates for which co-maintenance is possible: the latter occurs

in a narrow window in supplement fig. 12(a), but supplement fig. 12(b) then invokes an additional rate-

affinity trade-off which serves to widen the window of mutation rates for which co-maintenance is possible.

Supplement fig. 13(a) shows that a sigmoidal rate-yield trade-off alone is also sufficient to support multiple

quasispecies but the region of mutation rates for which co-maintenance is largest in these examples can be

seen found in supplement fig. 13(b). This illustrates that the width of the co-maintenance region depends

on both the nature and on the number of the trade-offs used, moreover supplement fig. 13(a) shows that the

number of types in the model may alter the size of the co-maintenance region.

The preceding numerical examples show that the theoretical principle of co-maintenance of quasispecies can

be realised in the MSC model and in the search for a rationale to support wider regions of coexistence, consider

the following example. Supplement fig. 14 illustrates that a yet wider region of mutation rates supporting

coexistence can be found, but in this figure we have invoked both rate-yield and rate-affinity trade-offs of the

form illustrated in supplement fig. 8(b). This lends credence to the argument that more complex, undulating

fitness landscapes stem from more complex trade-offs and that they, in turn, promote coexistence.

6.4.1. Co-maintenance of S. cerevisiae types from existing trade-off data. When known data for the rate-yield

and rate-affinity trade-offs estimated from the available literature for S. cerevisiae under glucose limitation

(see [45, 9] and supplement fig. 5) was deployed within the MSC model, we obtained the results shown in

supplement fig. 15. To produce this figure we used a washout rate of the chemostat of d = 0.375 per hour and

S0= 2500 mmol glucose/L, cells were measured in grams of biomass with cell density measured in grams of

biomass per litre (g/L). The uptake rate of glucose by a cell was then measured in units of mmol glucose per

gram of biomass per hour. We assumed for the purposes of this model that glucose uptake rates given in [45]

in fact represent maximal uptakes rates, a working assumption not necessarily true in practice but necessary

to permit us to formulate the data shown in supplement fig. 5 within the framework of the MSC model.

A comparison between the empirical data and how it has been processed for use within the MSC model is

given in supplement fig. 15(a), with the resulting fitness landscape illustrated in the same figure (rightmost

plot). Given these parameter values and prior empirical data, the MSC model predicts that co-maintenance

regions of both fast and efficient metabolic strategies can be found in the chemostat provided the mutation

rate lies within a region illustrated in supplement fig. 15(b). Although this is a narrow region, based on the

discussion of the previous section we anticipate that the inclusion of more biological detail into the model

would widen the region of mutation rates for which this co-maintenance property is found.

6.4.2. Are observed mutation rates consistent with the co-maintenance principle? Using our most realistically

parameterised model, namely the one illustrated in §6.4.1 above, from supplement fig. 15(b) we argue that

mutation rates at, or somewhat above an approximate threshold of ∼ 10−3.5= 0.0003 per genome per hour

would be sufficient to see the maintenance of both quasi-species.Assuming such an approximate figure

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION25

supplement fig. 15 – (a) The rate-yield and rate-affinity trade-off data estimated for S.

cerevisiae shown in supplement fig. 5 (repeated here as red circles) were interpolated and fitted

to a nonlinear model for deployment within the MSC equation. The result is a fitness landscape

with two peaks at high sugar concentrations (shown in the rightmost plot). (b) The resulting

steady-state solutions of the MSC equation (1) possess a region of co-maintenance of different

quasispecies at the appropriate mutation rates, the model also exhibits maintenance of the fit

and the flat at low and high mutation rates, respectively. A superposition of normal distributions

has been fitted to the type frequency histogram to highlight each quasispecies.

(a)

00.20.4 0.60.81

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

yield (g biomass /mmol glucose)

Rate−yield trade−off used in the MSC model

normalised uptake rate

Vmax = 20 mmol glucose /(g biomass x h)

00.2 0.40.60.81

10

20

30

40

50

60

70

80

90

Rate−affinity trade−off used in the MSC model

K (mmol/L)

normalised uptake rate

00.2 0.40.60.81

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fitness landscape

normalised maximal uptake rate

growth rate (h−1)

(b)

−3.8−3.6−3.4−3.2

60

70

80

90

100

110

120

130

140

150

log(ε)

density (g/L)

−3.8−3.6−3.4−3.2

0

0.5

1

log(ε)

multimodality index

00.2 0.40.60.81

0

0.5

1

S/S0 = 0.101, mutation rate (per cell per h) = 0.000633

normalised uptake rate

relative frequency

0 0.20.40.6 0.81

0

0.5

1

S/S0 = 0.093, mutation rate (per cell per h) = 0.000313

normalised uptake rate

relative frequency

00.2 0.40.6 0.81

0

0.5

1

S/S0 = 0.080, mutation rate (per cell per h) = 0.000212

normalised uptake rate

relative frequency

to be representative of more complex situations, and indeed of prokaryotes, we now ask whether observed

mutation rates are approximately of the correct order of magnitude to permit co-maintenance in practise.

From supplement fig. 15(b) it is seen that co-maintenance may be possible even as the mutation rate passes

beyond this threshold and so we ask whether it is likely that the experimental organism (E.coli in the case

of [21]) with the enigmatic recovery of mutiple types in a chemostat has a mutation rate that is sufficiently

high. Assuming E. coli exhibits about three generations per hour in a laboratory microcosm, the critical

mutation rate for such an organism would be of the order of 0.0003/3 = 0.0001 per genome per generation.

Most microorganisms, E. coli included, have a per genome mutation rate that is about an order of magnitude

higher than this at about 0.003 per genome per generation [7]. Only of a fraction of all the possible mutations

are likely to be relevant to the trade-offs, but if more than 0.0001/0.003 = 3.33% of the E. coli genome can

mutate and affect the relevant metabolism and physiology, then observed mutation rates are consistent with

our co-maintenance principle.

Any adjustment of these assumptions may affect this conclusion. For example, if we supposed that a

bacterial strain had an order-of-magnitude higher mutation rate, for example owing to stress or hypermutation

(as employed in [21]) with a genomic mutation rate of 0.03 per genome per generation, then just 0.3% of the

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26R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

genome would need to be involved. Similarly, if we permitted for each observable DNA-based mutation there

to be some rate of epimutation, again the fraction of sites involved in the trade-offs could be less than 3% for

the principle to apply.

6.5. Extension: asymmetric mutation matrices. Our analysis of the equilibria of the MSC equation

reveals that coexisting quasispecies can be maintained by negative-frequency dependence (NFD). Indeed,

the property of NFD follows a fortiori if a locally stable equilibrium of equation (1) possesses coexisting

quasispecies clouds, although the ecological reasoning that underpins the presence of NFD in this model will

not be discussed in any detail until §8 in this supplement.

If, for the moment, we accept the assumption of a stable, coexistence steady-state and therefore also the

presence of NFD in the MSC model, it follows that changing the mutational connectivity of types by altering

the mutation matrix M need not destroy the coexistence state and its local stability can be preserved under

such an alteration. As this will also maintain NFD, we claim that NFD may also be stable to the loss of

mutational connections between clouds. However, it is also true by the construction of the MSC model that

there must also be large changes to the connectivity encoded by M that will remove the quasispecies clouds

entirely. For example, removing all mutational connections is equivalent to setting ? = 0 and this will certainly

be sufficient to see the loss of coexistence.

A particularly important issue here concerns the bifurcation structures presented in Propositions 1 and 2

and how they change when we alter the mutation matrix M so that mutations between types are permitted

to occur asymmetrically between types. All the computations so far have utilised the mutation structure

illustrated in supplement fig. 2 extended to the appropriate number of types, it is therefore important that

we consider such asymmetries. The prior assumption of symmetry was invoked merely to shorten the proofs

of Propositions 1 and 2, one possible extension to cover certain asymmetric cases now follows.

Proposition 3. Suppose that (A1-A7) apply and that M is a symmetric, stochastic, irreducible n×n matrix

and A is any n × n matrix. Consider the following asymmetric perturbation of (7):

0=

?(M + A − I)b + (G(S) − d)b,

0=

d(S0− S) − ?U(S),b?.

For any strictly positive solution (b∗,S∗) of (17) defined when A = 0n×n, there is an α > 0 and a smoothly pa-

rameterised family of solutions of (17), (b(A),S(A)), defined for all ?A? < α such that (b(0n×n),S(0n×n)) =

(b∗,S∗).

(17a)

(17b)

Proof. Write the steady-state equation (17) using a smooth mapping in the form F(b,S;A) = 0 where we have

highlighted the fact that the perturbation of the mutation matrix M, namely A, is a parameter. Consider the

linearisation of F(b,S;A) with respect to (b,S) about any strictly positive non-trivial solution that, because

of Propositions 1 and 2, we know exists when A = 0n×nand that we now denote (b∗,S∗) for convenience:

−U(S∗)T

We now ask, can there be a vector (k,σ) ∈ Rn+1such that d(b,S)F(b∗,S∗;0n×n)[(k,σ)] = (0,0)? Writing this

linear equation in terms of its two components we require

d(b,S)F(b∗,S∗;0n×n) =

?(M − I) + diag(G(S∗)) − dIG?(S∗)b∗

−(d + ?U?(S∗),b∗?)

.

(?(M − I) + diag(G(S∗)) − dI)k + σG?(S∗)b∗

−?U(S∗),k? − σ(d + ?U?(S∗),b∗?

=

0

(18a)

=0,

(18b)

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION27

where G?(S) denotes the derivative of G(S) with respect to S, this is a strictly positive vector by earlier

assumptions. Taking the inner product of (18a) with the strictly positive vector b∗and using the symmetry of

M (we assumed M = MT) we find σ·?b∗,G?(S∗)b∗? = 0 from where σ = 0 must follow. From equation (18a)

we then deduce that k = θ · b∗for some real θ so that ?U(S∗),θ · b∗? = 0 must hold by (18b), it therefore

follows that θ = 0.

As (k,σ) must be the zero vector, the derivative d(b,S)F(b∗,S∗;0n×n) is an isomorphism and so we can

apply the implicit function theorem to deduce the existence of a locally-defined path of solutions of (17) that

depends smoothly on A:

F(b(A),S(A);A) ≡ 0

such that

b(0n×n) = b∗,S(0n×n) = S∗.

According to the implicit function theorem, the path (b(A),S(A)) is smoothly-defined on some neighbourhood

of the matrix 0n×n, as claimed.

?

supplement fig. 16 – Using a reducible mutation matrix in (1) can lead to hysteresis:

increasing the mutation rate creates two fold bifurcations and three steady-states, of which two

are asymptotically stable and one is unstable. Here, the fit types (red) give way to types on

flatter peaks (blue) as the mutation rate increases.

−4.4−4.2−4 −3.8−3.6

6.1

6.15

6.2

6.25

6.3

6.35

6.4

6.45

log10(ε)

log10 density

Steady−state densities

0 0.2 0.40.6 0.81

0

0.5

1

S/S0 = 0.31845 , mutation rate = 7.7872e−05

normalised uptake rate

S/S0 = 0.31566 , mutation rate = 7.986e−05

relative frequency

0 0.20.40.60.81

0

0.5

1

normalised uptake rate

relative frequency

Proposition 3 states that whenever we find a steady-state solution of (1), introducing asymmetric mutation

rates between different cell types will not change the structure of the steady-state discontinuously. So, if we

find the co-maintenance of fit and flat cell types in some system, that co-maintenance property will persist

even if we alter the mutation matrices to be asymmetric. However, we re-iterate that there are certain

symmetric perturbations of the mutation matrix M that can destroy coexistence and, as a result, there are

also asymmetric perturbations that will have the same effect.

Remark 3. Do note that the set of restrictions (A1-A7) we have imposed on the MSC model must be taken

as sufficient conditions leading to the coexistence hypothesis at the appropriate mutation rates. They are

not necessary conditions, nor do we claim that all models of the form (1) have such a coexistence region.

Indeed, modifications of our underlying assumptions could result in the destruction of the coexistence region,

in particular the existence of more than one steady-state can lead to hysteresis in the bifurcation structure (see

supplement fig. 16) and under these conditions our coexistence hypothesis, as with other quasispecies models

[37], may well be false.

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28R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

7. An individual-based, stochastic model

In order to test the predictions obtained using the MSC model (3) and presented in the previous section,

we now develop an individual-based, stochastic chemostat model into which we can encode rate-yield and

rate-affinity trade-offs within single cells in competition for one limiting resource. Rather than develop a

version of the MSC model as a stochastic ODE, we quite purposefully create a rule-based system that we

know cannot be well-described in every detail by the MSC equations. For example, while the mutation rate

was defined per unit time in the MSC model, mutations will be assumed to occur per division for the model

that we describe below. Our rationale for this is the following. If the predictions of the previous section are

to have any broad-scale relevance, we ought to be able to see those predictions borne out in quite different

theoretical model systems.

The stochastic model, completely absent of any spatial structure, is encoded by the following rules that are

evaluated each computational generation:

(SM1). There are at most n different possible cell types where each type is labelled by a number i between 1

and n. A cell of type i has that number of surface transporters for the resource molecules and each

transporter can only contain one molecule at a time. As a result, the per unit time maximal uptake

of resource by each cell, its Vmax, coincides with its type.

(SM3). Free resource molecules, finite in number (denoted S) are associated and bound randomly to each of

the cells’ multiple transporters such that the probability of binding is S/(Ki+S) per transporter per

unit time; here, the phenotype Kiis a half-saturation constant associated to each cell type. Bound

transporters port molecules immediately across the cell membrane which are then held within the cell

at no energy cost.

(SM3). The cell is assumed to convert internally held resource into ‘protein’ and when sufficient protein has

accumulated fission is triggered, at this step a cell of type i is assumed to require at least eimolecules

of resource before cell division can occur.

(SM4). Mutation rate is measured per cell division: when a cell divides, a copy is made with a fixed probability

that the cell’s offspring has type i + 1 or i − 1, given a parent cell of type i. Mutations that occur

when a parent with types 1 and n divides can only yield offspring of types 2 and n − 1, respectively.

(SM5). When a cell divides, the internally held resources of the parent are apportioned symmetrically to

create a parent and offspring pair with identical internal states.

(SM6). At each computational generation, d% of all biotic and abiotic matter is lost from the chemostat and

resource molecules are supplied to the device at a constant rate.

As i is the maximal uptake rate of cell type i, a rate-yield trade-off can be incorporated into this stochastic

model by having eiincrease as a function of i, noting that 1/eiis the cell yield per unit resource. A rate-

affinity trade-off is encoded in the stochastic model by having Kiincrease as a function of i and the trade-offs

used throughout this section, unless otherwise stated, are shown in supplement fig. 17. For the washout

parameter we used the value d = 2%, unless stated otherwise, so that the mean length of stay of any particle

in the simulated chemostat device is fifty computational generations and the resource supply concentration

was chosen to yield a carrying capacity of approximately three thousand cells in steady-state. These values

were chosen for the convenience of rapidly simulating the model in Matlab and we chose a number n of fifty

types throughout.

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION29

supplement fig. 17 – The rate-yield trade-off and the rate-affinity trade-offs used in the

stochastic model: this is analogous to supplement fig. 8(b), adapted to the units of the stochastic

model.

0.20.40.60.81

0.14

0.15

0.16

0.17

0.18

0.19

yield (1/resouces needed per division)

Rate−yield trade−off in the stochastic model

normalised uptake rate

Vmax = 50 molecules per cell p.u.t

0.20.40.6 0.81

100

150

200

250

300

K (molecules)

normalised uptake rate

Rate−affinity trade−off in the stochastic model

7.1. Neutrality in the stochastic model. The vectors (ci) and (Ki) that associate cell types with uptake

rate, cell yields and resource affinities S determine the nature of the fitness landscape, along with the resource

concentration S. If (ci) and (Ki) are constant across cell types then the cell fitnesses form a neutral landscape

and so their frequencies change through drift alone, although the total density of the population can change

in time.

The long-term dynamics in this neutral scenario are illustrated in supplement fig. 18 when there are just

two types, so n = 2, where the washout rate is d = 10% and the mutation rate was set to zero for the

production of this figure. For these parameter values, one type is eventually lost with probability one and

the between-population invariant density of the stochastic model is then a weighted sum of Dirac masses

supported on each of the two types. This claim is consistent with the histogram depicted in supplement

fig. 18 formed from a finite number of observations.

supplement fig. 18 – The neutral stochastic model with n = 2 cell types: in each realisation

of the model, the fixation of one of the two types occurs eventually with probability one.

Averaging between populations yields an estimate of the invariant density, a weighted sum of

two Dirac masses supported at zero and one:

1

2(δ(·) + δ(· − 1)).

00.20.40.6 0.81

0

0.1

0.2

0.3

0.4

0.5

Type histogram: neutral model with 2 types

Type 1 frequency at end of simulation

Frequency observed

1020 observations

7.2. Competitive exclusion in the stochastic model. Consider the non-neutral fitness landscape pictured

in supplement fig. 14(a), itself derived from the rate-yield and rate-affinity trade-offs shown in supplement

fig. 7. When we previously employed this set of trade-offs in the MSC model we obtained the cell type densities

depicted in supplement fig. 14(c). This figure illustrates that there is a wide range of mutation rates for which

two quasispecies can coexist with approximate normalised uptake rates of 0.2 and 0.6 units, when taking into

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30R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

account the 50 cell types of the stochastic model, the mode of the coexisting fit and flat quasispecies should

therefore be located near to cell types 10 = 0.2 × 50 and 30 = 0.6 × 50.

We then ran the stochastic model for 40,001 computational generations with the mutation rate set to zero,

as shown in supplement fig. 19(a), and this established type 11 as the fittest under the prevailing model

parameters. Furthermore, supplement fig. 19(b) shows the outcome of a direct competition between type 11

and the only other type that also persisted to the end of the computation in 1,000 replicates, namely type

31. Again, type 11 prevailed in all replicates, just ten replicate trajectories are shown as an illustration in

supplement fig. 19(b).

These computations verify that the stochastic model is consistent with survival of the fittest when mutation

rate is set to zero and indicates that type 11 is the fittest cell type for the parameters used. Based on

the analysis and computations of the previous section, we therefore anticipate that a cloud of quasispecies

containing type 11 will coexist with different quasispecies at sufficiently high mutation rates.

supplement fig. 19 – With selection acting through both rate-yield and rate-affinity trade-

offs but zero mutation rate, we anticipate competitive exclusion. (a) This plot of persistence

times for each cell type shows that one emerges as the fittest. (After 40,001 computational

generations and of 50 possible types, type 11 fixed in 898/1,000 replicates and was the only

type never to be lost from the population. It also had the highest density by the end of every

simulation; type 31 was the only other type that persisted to the final computational generation

of any simulation.) (b) Competition between just two types, 11 and 31, indicates 11 to be the

fitter (shown are ten superimposed replicates and their means, each lasting 2,000 computational

generations with d = 10%).

(a)

01234

x 10

4

5

10

15

20

25

30

35

40

45

50

40001

25th−75th %−iles

mean

time (computational generations)

type

Persistence times of all cell types (mutation rate = 0)

40001 generations, uniformly distributed initial data, 1000 replicates

(b)

20406080100 120140160180 200

0

100

200

300

400

500

600

time (x 10)

cell density

Competitive exclusion holds between any two cloud members

type 11

type 31

7.3. Co-maintenance of the fit and the flat in the stochastic model. Having established consistency of

the stochastic model with prior neutral and competitive systems under the appropriate model restrictions, in

order to estimate equilibrium densities and type frequencies of this model we computed mean type distributions

both within and between different populations at different mutation rates. The results, shown in supplement

fig. 20 for a range of mutation rates, are based on means taken between two hundred different populations, all

differing in the initial distribution of types and where the initial allocation of internal resources was randomly

distributed among the 3,000 cells.

It is clear from supplement fig. 20 that a bimodal distribution of cell types is obtained at sufficiently high

mutation rates whereas a near-monomorphic distribution in mutation-selection balance is obtained at the

lowest mutation rates. We used a multimodality index defined as the ratio of total densities found in the two

observed quasispecies to further probe the onset of coexistence. As can be seen in supplement fig. 20(bottom-

right), before the onset of coexistence the multimodality index resides near zero, however fluctuations in this

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION31

supplement fig. 20 – A plot of the long-term densities of cell types measured between

populations using the stochastic model for increasing mutation rates and indicating the presence

of two quasispecies for sufficiently high mutation rates. The very bottom-right diagram shows

the ratio of mean biomass under each cloud and indicates that a critical mutation rate must be

exceeded for two clouds to coexist. (Histograms of between-population means of type densities

were computed using two hundred randomly generated initial populations; standard errors are

indicated.)

0 1020 3040 50

0

1000

2000

3000

mutation rate = 0

mean density

type

01020304050

0

500

1000

1500

2000

mutation rate = 0.016

type

0102030 4050

0

500

1000

1500

mutation rate = 0.031

type

01020304050

0

500

1000

1500

mutation rate = 0.062

mean density

type

0 10203040 50

0

500

1000

mutation rate = 0.12

type

01020304050

0

200

400

600

800

mutation rate = 0.25

type

01020304050

0

200

400

600

mutation rate = 0.5

mean density

type

01020304050

0

200

400

600

mutation rate = 1

type

−Inf−6 −5 −4 −3 −2 −1

log2(mutation rate)

0

0

0.1

0.2

0.3

multimodality index

measure become more apparent as the flatter quasispecies persists for larger times as the transition is neared.

Once a critical mutation rate is passed, variability of the multimodality index initially reduces but the index

itself increases sharply and two quasispecies are co-maintained, one containing cell type 11 and the other type

31.

In order to establish whether the bimodality of between-population means shown in supplement fig. 20 was

found within each population, samples were taken of temporal trajectories of the stochastic model for fixed

initial conditions with a single seeding type (chosen to lie at the mid-point of type space {1,2,...,50}, namely

25). The individual states of each of the 3,000 seeding cells were again randomly chosen, where the latter value

was chosen to be approximately equal to the carrying capacity of the simulation and the mutation probability

per division was set to 1/8. A typical result is shown in supplement fig. 21 with similar results obtained

for different mutation rates between 1/8 and 1 (data not shown). Diagram (a) of supplement fig. 21 shows

a simulated phylogeny obtained after 20,001 computational generations in which the seeding monomorphic

population branches into three lineages of which one is eventually purged.

The mean type densities associated with the final 2,000 generations of the single population depicted in

supplement fig. 21(a) are shown in supplement fig. 21(b,left), this is a bimodal distribution and satisfies the

Hartigan dip test for bimodality (p < 10−14). To check that such bimodality was maintained for all large

enough times, the lower and upper envelopes of the same trajectory are superimposed as blue lines on the

diagram and a histogram of type densities taken between 200 populations for the same initial states is shown

in supplement fig. 21(b) for comparison. The upper and lower envelopes are defined to be the maximal and

minimal values taken by type densities throughout the simulated trajectory and so provide upper and lower

bounds on that trajectory. So, if the data ensemble of length N = 2,000 type densities on n = 50 types, is

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32R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

supplement fig. 21 – (a) An initially monomorphic population branches into three qua-

sispecies of which one is eventually lost (simulation of 20,001 computational generations). (b)

The long-term cell type densities found within the population in (a) (shown as a black line) are

compared with long-term averages computed between populations for the same mutation rate

(blue histogram). Also shown (blue lines) are the upper and lower envelopes of the final 1,000

computational generations from the phylogeny in (a) illustrating that the bimodality of efficient

and inefficient types is maintained within one population for all sufficiently large times.

(a)

6

7

8

9

10

11

12

13

14

27

28

29

30

31

32

33

34

35

36

37

38

25

evolutionary time

type

Simulated phylogeny with 50 possible cell types

(mutation rate 0.125 per cell per division)

(b)

1020304050

0

200

400

600

800

1000

type

densities

Within−population densities (mutation rate 0.125)

maximal envelope

minimal envelope

mean (2000 comp. gens)

1020304050

0

200

400

600

800

1000

type

densities

Between populations

mean (200 populations)

E = (∆(1)

t ,∆(2)

t ,...,∆(n)

t

)N

t=1then the upper and lower envelopes are defined to be the vectors

?

?

Both of these are bimodal, as can be seen in supplement fig. 21(b,left), we therefore deduce that the type

distribution contains two clouds or clusters for each of the 2,000 computational generations sampled.

upp(E) =max

1≤t≤N∆(1)

t , max

1≤t≤N∆(2)

t ,..., max

1≤t≤N∆(n)

t

?

?

and

low(E) =min

1≤t≤N∆(1)

t , min

1≤t≤N∆(2)

t ,..., min

1≤t≤N∆(n)

t

.

7.4. Preventing valley-crossing lineages: mutational barriers. As the phylogeny in supplement fig. 21(a)

indicates, the observed bimodality of the long-term type distributions consisting of efficient and inefficient cell

types in supplement fig. 21(b) does not appear to be due to lineages traversing fitness valleys from the fit

to the flat and so replenishing the flatter quasispecies. In order to test this hypothesis more thoroughly, we

performed two further tests.

First, after executing a single simulation of the stochastic model and running it to a near-equilibrium state

(based on the mean type distributions shown in supplement fig. 20) we then imposed a mutational barrier

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION33

supplement fig. 22 – The presence of two quasispecies persists within populations even after

the imposition of mutational barriers between quasispecies. Comparing the black line which is

the between-population type distributions taken from supplement fig. 20 with the resulting

within-population distributions obtained after a mutational barrier has been imposed in the

stochastic model, diagram (b) shows little change, (a) shows the loss of the flattest cloud and

(c) shows a small increase in the density of some of the flat types. (In all three diagrams, the

mutation rate has been set equal to its highest possible value, namely 1 per cell per division.)

(a)

510152025

type

3035 404550

0

100

200

300

400

500

600

700

800

900

mean density

Mean cell densities with and without mutational barriers

barrier at 13 (mean of 5000000 generations)

no barrier (mean of 2200 generations)

(b)

510152025

type

3035404550

0

100

200

300

400

500

600

mean density

Mean cell densities with and without mutational barriers

barrier at 23 (mean of 200000 generations)

no barrier (mean of 3600 generations)

(c)

510 152025

type

30 35404550

0

100

200

300

400

500

600

mean density

Mean cell densities with and without mutational barriers

barrier at 28 (mean of 200000 generations)

no barrier (mean of 4600 generations)

supplement fig. 23 – (a) Diversity of quasispecies is maintained at all times by negative

frequency dependence (fifty tests, mutation rate equals 1) even in the presence of a mutational

barrier between those quasispecies (at type 23).(b) This is illustrated by long-term time

averages within a single population trajectory and in the lower and upper envelopes of that

trajectory (after transients) all of which illustrate the persistence of both quasispecies. For

comparison, the right-hand diagram shows between-population averages (taken from supplement

fig. 20) with no mutational barrier.

(a)

0 0.20.40.60.81

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

initial frequency of flat cloud

relative change in frequency of flat cloud

Change in frequency of flat cloud (50 tests)

(b)

01020304050

0

100

200

300

400

500

600

type

densities

Within population densities (mean of 2000 comp. gens.)

(mutation rate = 1 and barrier at type 23)

maximal envelope

minimal envelope

mean

01020304050

0

100

200

300

400

500

600

type

densities

Between populations, no barrier

(mutation rate = 1, no barrier)

between quasispecies and observed changes in the long-term distribution of type densities that resulted from

this. A mutational barrier in a linear mutational topology is a pair of cell types between which the probability

of mutation is zero, see supplement fig. 2(b) for an illustration, such a mutational structure is asymmetrical in

the sense of §6.5. The consequences of imposing a barrier are shown in supplement fig. 22 where two clouds of

types may be found even after the imposition of a barrier, although supplement fig. 22(c) shows that barriers

can enhance the densities of one quasispecies and supplement fig. 22(a) shows that one of the quasispecies

may be lost after a barrier is introduced.

Second, we tested the stochastic model for negative frequency dependence (NFD) at the level of quasispecies

by imposing mutational barriers between clusters and then suppressing the densities of each of the two

emergent quasispecies clusters in turn. The resulting change in frequencies of the types in time within each

cluster, whether increasing (a relative change in frequency larger than one) or decreasing (a relative change in

frequency less than one) is shown in supplement fig. 23(a) for fifty replicates of one initial condition supporting

two quasispecies. Our point is that this figure clearly supports the coexistence hypothesis through negative

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34R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

frequency-dependent selection. Moreover, the mean, lower and upper envelopes of the simulated distribution

of types of just of one of these replicates is shown in supplement fig. 23(b,left) illustrating that bimodality of

the distribution of types was maintained at all times.

7.5. Robustness to changes in mutation kernel. Let us momentarily modify the nature of mutations in

the stochastic model in the following manner, changing (SM4) above to be the following rule:

(SM4?). When a cell divides, a copy is made but with a fixed probability that the cell’s offspring has a different

type to its parent. If w is a fixed parameter, the type of the offspring is a randomly chosen, uniformly

distributed member of the set {i−w,...,i−1,i+1,...,i+w} given a parent cell of type i. Mutations

that occur when a parent with types 1 and n divides can only yield offspring of types {2,...,w} and

{n − w,...,n − 1}, respectively.

When using assumption (SM4?) we say that the mutation kernel is the uniform distribution of width w.

supplement fig. 24 – The co-maintenance of two quasispecies is robust to changes in mu-

tational structure, illustrated here as the width w of a uniformly-distributed mutation kernel

is increased. In the extreme, the complete absence of correlation between parent and offspring

types leads to the loss of both quasispecies (bottom-right figure). (Shown are histograms of

cell type densities obtained after a transient period and averaged over a trajectory of 2,000

computational generations.)

0 1020304050

0

200

400

600

800

type

densities

Mutation kernel of width 1

0 10 2030 4050

0

200

400

600

800

type

densities

Mutation kernel of width 2

01020 304050

0

200

400

600

800

type

densities

Mutation kernel of width 3

01020 304050

0

200

400

600

800

type

densities

Mutation kernel of width 4

01020304050

0

200

400

600

800

type

densities

Mutation kernel of width 5

0102030 40 50

0

200

400

600

800

type

densities

No parent−offspring correlation

(mutation rate = 0.25)

(30000 transient generations)

(mean of 2000 generations )

Supplement fig. 24 shows the density of types obtained when simulating the stochastic model after making

the choice of w = 1,2,3,4 and 5, thus increasing the width of the mutation kernel. The same figure also

illustrates the density of types when choosing each mutated offspring type randomly from the set {1,2,...,n},

in this case we say that there is no parent-offspring correlation of types. The purpose of supplement fig. 24 is

to show that a population structure consisting of co-maintained quasispecies is robust to changes in mutation

structure but that a correlation between parent and offspring must be present in order that the quasispecies

are stably maintained.

8. Co-maintenance: a toy model

To understand what might be the underlying cause of the maintenance of different quasispecies by negative-

frequency dependent selection, consider the following reasoning. Returning to (1), note that the model is

subject to both frequency and density-dependent selection. However, when the mutation rate ? is small a

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION35

mathematical argument based on the implicit function theorem (and used in Proposition 1) tells us that

one type, the fittest, dominates and all others are small in frequency. More precisely, up to a re-ordering of

coordinates the steady-state vector f = (f1,f2,...,fn) of (1) is in a state of mutation-selection balance in the

sense that there is an ?-independent constant κ > 0 such that

1 − κ? < f1≤ 1 and0 ≤ fj< κ? for j ≥ 2,

for all small enough ?.

Once a critical mutation rate is surpassed, a second quasispecies can be supported even in the presence

of a mutational barrier because equation (1a) tells us that one cloud can mediate the dynamics of the other

through their common environment, S. To see this, imagine the thought experiment whereby we run the

chemostat to steady-state conditions and isolate two quasispecies, one fit and one flat, and we then perform

the following exchange of cells. (1) First remove one cell from the flat cloud and then add one cell to the fit

cloud and observe the dynamics of the system, (2) in a second population remove one cell from the fit cloud

and add one cell to the flat. How does the system return to its equilibrium state?

supplement fig. 25 – (a) The standard test of tracking the response in frequencies of

cell types demonstrates that two quasispecies are supported by negative frequency-dependent

selection in the MSC model (1). (b) Response of the resource concentration after increasing

the frequency of fit types (left) and flat types (right): the former case leads to a momentary

decrease in sugar concentration, the latter to a momentary increase. This perturbation of

the environment sets the conditions for both quasispecies and the resource concentrations to

return to equilibrium (shown as red lines). (The trade-offs used in (a) and (b) are illustrated in

supplement fig. 8(b).)

(a)

00.20.4 0.60.81

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

initial frequency of fit cloud

relative change in frequency of fit cloud

Testing the MSC model for NFD (10 tests)

(b)

0510 1520

0

0.2

0.4

0.6

0.8

1

time (hours)

sugar concentration S(t) (units of S0)

Sugar dynamics after altering cloud frequencies

steady−state

increased high−uptake types

increased high−yield types

The answer lies in the fact that these two operations momentarily affect the dynamics of the sugar in the

environment in different ways, as depicted in supplement fig. 25(b). In the case where the frequency of fit

types is increased, note that these types are fitter because of their higher uptake rates, hence the resource

concentration reduces from its previous steady-state value. However, at such lower resource concentrations it

is the ‘flatter cells’ with their higher efficiency that are advantaged by this change which then acts to increase

and so re-set sugar levels back towards their equilibrium value. The case where the flat types are increased in

frequency behaves entirely analogously, but by first increasing sugar levels (see supplement fig. 25(b)).

When this thought experiment is implemented in simulations of (1), the negative frequency-dependent

dynamics that result are shown in supplement fig. 25(a) whereby the fit and flat types are both returned to

intermediate frequencies when rare or common.

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36R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

8.1. A toy population genetical model. Having established the cause of environmentally-mediated, nega-

tive frequency-dependent selection, we now continue our intuitive explanation regarding mechanisms enabling

coexistence of the fittest and the flattest by developing a toy model. So, consider two quasispecies each labeled

a and A formed from two distinct clusters of types. The fitness of quasispecies a is denoted by Gawhile the

fitness of quasispecies A is denoted by GA. The relative frequency of the set of all types in quasispecies A is

given by fAand we assume that negative frequency dependence is exhibited by this model in the sense that:

(19)

GA

Ga

= r0− fA(r0− r1)

where r0 > r1 > 1 are fixed constants and, by assumption, 0 ≤ fA ≤ 1. These inequalities imply that

GA> Ga, meaning A is the fitter of the two quasispecies.

Next, we assume that there are three types of mutational event that occur between quasispecies that arise

due to the mutation of different cell types within them:

(ME1). a → a or A → A: these cases correspond to mutational events that keep offspring cell types within

the same quasispecies as their parent type (this is an intra-quasispecies mutation);

(ME2). a → A or A → a: these cases correspond to mutational events that move offspring cell types to

different quasispecies from their parent types (an inter-quasispecies mutation);

(ME3). a → dead or A →dead: here ‘dead’ represents a highly-deleterious mutation that produces a non-viable

offspring cell type neither in quasispecies a nor A.

To discount mutation-selection equilibrium we assume that mutational events of type (ME2) do not occur,

moreover the fitness effects of mutational events of type (ME1) are completely ignored in this simple model.

We now suppose that the loss of cell types from a and A due to a mutational event (ME3) reduces the

fitness of a and A in the sense that

(20)

wa= Ga− µa

and

wA= GA− µA,

where µaand µA> 0; thus waand wAare the fitnesses of each quasispecies in the presence of a constant

per-type, per-cell-division mutation rate. We shall assume that µA> µa, expressing the idea that a is the

flatter quasispecies and the more mutationally robust, it follows that mutations that occur to cell types within

quasispecies a are less likely to be non-viable. We now assume for reasons of simplicity that there is an α

such that

(21)

α = µa/µA

with 0 < α < 1 from our construction.

Now, the principle of survival of the fittest is relevant when wA> waand a little algebra shows that this

occurs when the following inequality applies:

1−α

relevant when wA< waand, again, a little algebra shows that this occurs if

model so-constructed is consistent with the survival of the fittest and flattest at low and high mutation rates,

respectively. Coexistence of the fittest and the flattest quasispecies is relevant when the equality wA= wa

holds, based on our construction we find that this occurs when mutation rates take on intermediate values in

such a way that

Ga(r1− 1)

1 − α

see supplement fig. 26(a) for an illustration.

This co-maintenance principle requires two core properties, namely GA> Ga,µA> µaand that negative

frequency-dependent selection acts between the two quasispecies. If we were to assume that mutations to

Ga(r1−1)

> µA. On the other hand, survival of the flattest is

Ga(r0−1)

1−α

< µA. Thus, our toy

< µA<Ga(r0− 1)

1 − α

;

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THE FITTEST AND THE FLATTEST: SUPPLEMENTARY INFORMATION37

types in quasispecies a have the same probability of being non-viable as mutations to types in quasispecies

A, in the sense that µA= µa, it would then follow that A is a fitter quasispecies than a. Similarly, suppose

there were an absence of negative frequency dependence, r0= r1, in this instance (19) would become

(22)

GA= r0Ga.

Allowing for µa= αµAwith 0 < α < 1 and substituting (22) into (20) we obtain

(23)

wA

wa

=Gar0− µA

Ga− αµA.

From (23) we deduce that there could only be a single value of µAfor which coexistence of the fittest and the

flattest could occur, namely for µA=Ga(r0−1)

1−α

possible to derive the non-generic or non-robust form of coexistence depicted in supplement fig. 26(b).

and thus, in the absence of these two core properties, it is only

supplement fig. 26 – The relative fitness of quasispecies ‘A’ as a function of the frequency

of all the types contained within it, fA. (a) The toy population genetical models predicts that

the transition of the fittest quasispecies ‘A’ (see label (1)), to the flattest ‘a’ (see label (3))

passes through a regime of co-maintenance (see label 2) in the direction indicated by the arrow

as the mutation rate is increased. (b) Only a non-robust form of co-maintenance (label (2)) is

possible in the absence of negative frequency-dependence.

(a)

1

1

0

(1)

(2)

(3)

(b)

1

1

0

(1)

(2)

(3)

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Appendix A. Model parameters

The parameter values employed for the MSC model for all figures apart from supplement fig. 15 were

adapted from ones used previously by the authors [10] in the study of E. coli B under glucose limitation. Cell

concentration is given in cells/ml, values of Vmaxand resource uptake by each cell are given in µg per cell

per unit time, the chemostat washout parameter d is measured per hour so growth and uptake rates are also

measured per hour. Cell yield is given in cells per µg of resource and resource concentrations for S0are stated

in µg/ml, S is then stated as number between 0 and 1 as a fraction of S0. The precise parameter values

used cannot be presented succinctly due to the trade-offs used, but throughout d was taken between 0.2 and

1.4h−1, S0was 1µg/ml, K between 0.01 and 0.1µg/ml, Vmax≈ 10−6µg/cell/h with cell yield approximately

106cells/µg. As a result, growth rates of all cell types range between 0 and 2 per hour, with most types around

one division per hour. This gives equilibrium cell densities in the culture vessel of the chemostat between 106

and 108cells per ml, with lower densities generally found at higher mutation rates.

Different physical units were used for the implementation of the stochastic model. The large concentration

of cells found in chemostat devices prevents the use of parameters in the same units as the MSC model because

of constraints on computational resources. So, parameters were used to yield a carrying capacity of 3,000 cells,

a number that could be simulated rapidly to generate enough replicates to ensure reliable statistics. Available

resources are measured as a number of molecules in the chemostat device but volume is not specified, cells

are given as a number, the unit of time is arbitrary and measured according to iteration number of the model

(we call this a computational generation). Unless stated otherwise, washout is 2% per iteration yielding an

expected time to loss from the device of 50 computational generations for every cell and resource molecule.

Cell growth is modelled by having each cell accumulate a minimal amount of protein in order to divide and

this provides an efficiency measure because it is assumed that each resource molecule can be converted into

a fixed and constant amount of protein. All cells have the same mutation rate which is the probability of a

change in type per division. A mutation changes the type of a cell by a value of at most w to ensure a positive

correlation between the type of a parent cell and the type of the offsprings derived from it. This assumption

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40R. E. BEARDMORE, I. GUDELJ, D. A. LIPSON AND L. D. HURST

was purposefully invoked to prevent mutations in a parent cell of one quasispecies or fitness peak that could

yield offspring cells in a different quasispecies or fitness peak.

Supplement figure 15(b) is based on known Saccharomyces cerevisiae data and was produced using the rate-

yield and rate-affinity data provided in supplement fig. 15(a) where a chemostat washout rate of d = 0.375 per

hour was used, with a glucose supply rate of S0= 2500 mmol/L. Glucose uptake here is measured in mmol

per gram of biomass per hour with a maximum possible value of Vmax= 20 mmol/g biomass/h, this is the

maximum value for which data is available in supplement fig. 5(c). As can be seen in supplement fig. 15(a),

this yields a prediction of between 70g and 140g of biomass per litre.

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