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MAINTENANCE OF THE FITTEST AND THE FLATTEST

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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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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