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Mathematics 2023, 11, 2473. https://doi.org/10.3390/math11112473 www.mdpi.com/journal/mathematics
Article
Summation Laws in Control of Biochemical Systems
Hans V. Westerhoff
1,2,3,4
1
Department of Molecular Cell Biology, Amsterdam Institute for Molecules, Medicines and Systems,
Vrije Universiteit Amsterdam, De Boelelaan 1108, 1081 HZ Amsterdam, The Netherlands;
hvwesterhoff@gmail.com
2
School of Biological Sciences, Faculty of Biology, Medicine and Health,
The University of Manchester, Manchester M13 9PT, UK
3
Swammerdam Institute for Life Sciences, University of Amsterdam, Sciencepark 904,
1098 XH Amsterdam, The Netherlands
4
Stellenbosch Institute of Advanced Studies (STIAS), Wallenberg Research Centre at Stellenbosch University,
Stellenbosch 7600, South Africa
Abstract: Dynamic variables in the non-equilibrium systems of life are determined by catalytic ac-
tivities. These relate to the expression of the genome. The extent to which such a variable depends
on the catalytic activity defined by a gene has become more and more important in view of the
possibilities to modulate gene expression or intervene with enzyme function through the use of
medicinal drugs. With all the complexity of cellular systems biology, there are still some very simple
principles that guide the control of variables such as fluxes, concentrations, and half-times. Using
time-unit invariance we here derive a multitude of laws governing the sums of the control coeffi-
cients that quantify the control of multiple variables by all the catalytic activities. We show that the
sum of the control coefficients of any dynamic variable over all catalytic activities is determined by
the control of the same property by time. When the variable is at a maximum, minimum or steady,
this limits the sums to simple integers, such as 0, −1, 1, and −2, depending on the variable under
consideration. Some of the implications for biological control are discussed as is the dependence of
these results on the precise definition of control.
Keywords: control coefficients; metabolic control analysis; systems biology; genomics;
pharmacokinetic principles; systems biology and PBPK; time-dependent control analysis; systems
pharmacology; growth rate; yield and efficiency
MSC: 92B99
1. Introduction
The chemical networks in living organisms are all organized in the same hierarchy:
a genome contains genes for proteins. Many of these are enzymes, i.e., catalysts of chem-
ical (and transport) reactions in metabolism. The expression of a gene may be affected by
addressing the DNA region upstream, by mutation or transcription factor. The effect is an
altered level of the corresponding enzyme. At time scales that are important for physiol-
ogy, i.e., function, intracellular metabolism is at a quasi-steady state. Hereby the metabo-
lite concentrations and the fluxes become functions of the enzyme concentrations and
hence of the gene activities. This organization has implications for the mathematics of the
behavior of living cells, as we shall elaborate on below.
The sequencing of whole genomes, with the subsequent mapping of the roles that
the subset of metabolic genes play in human metabolism [1], has led to an increased at-
tention on how biological function may be adjusted by modulating gene expression or
enzyme activities. With his background in both genetics and theoretical biochemistry,
Citation: Westerhoff, H.V.
Summation Laws in Control
of Biochemical Systems.
M
athematics 2023, 11, 2473.
https://doi.org/10.3390/math1111247
3
Academic Editors: Yaroslav
Nartsissov, Jianjun Paul Tian and
Takashi Suzuki
Received: 25 March 2023
Revised: 4 May 2023
Accepted: 26 May 2023
Published: 27 May 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/license
s/by/4.0/).
Mathematics 2023, 11, 2473 2 of 11
Henrik Kacser, together with James Burns, realized how gene copy number could deter-
mine biological function quantitatively. They identified the molecular basis of dominance
[2]. The explanation is that the sum over all the enzymes in a network, of their control
coefficients on a flux, must equal one [3,4], and that there are many enzymes in the usual
networks, so that the average control must be small (1/n for n enzymes in a linear pathway
[2]). The fact that the sum of the flux control coefficients must equal one at steady state
inspired Kacser and others to ask whether that control would then be distributed homo-
geneously among the enzymes or reside in a single key factor. For most biochemical net-
works, the answer appears to be: distributed but not such that all enzymes have the same
small control on that flux, e.g., [5–13].
The importance of these results cannot be underestimated. For many years the concept
that each metabolic pathway should have a single ‘rate-limiting’ step, preferably at its begin-
ning, dominated both biochemistry and molecular biology [14–18]. Based on this concept, re-
searchers searched for such rate-limiting enzymes, key genes, and key regulators, without
finding these unequivocally. The number of oncogenes for instance, is not equal to 1, which
would have been in accordance with the concept, but >100, with dire consequences for diag-
nosis and therapy [19]. Only relatively recently, has biomedical research begun focusing on
drug combinations for the therapy of multifactorial diseases such as cancer, type-2 diabetes,
and heart failure [20–22]. For microbiology the realization that the specific growth rate of mi-
croorganisms may be limited by multiple factors at the same time is of great importance for
the development of new antibiotics in the context of multidrug resistance; and likewise for
cancer [23,24].
At first, the summation laws were limited to metabolic fluxes and concentrations at
steady state, with 1 and 0 for their respective sums. Subsequently, the laws have been
extended to include time dependent metabolite concentrations and fluxes [25], as well as
the time dependencies themselves [26,27]. Proofs of these laws were based on implicit
differentiation [4,28], or on the theory of homogeneous functions [29,30]. The properties
for which the laws have been derived remain limited in number. They do not include the
specific growth rate, growth yield, and thermodynamic efficiency that are important for
microbiology, for instance. Nor do they address the control of the transformation status
of a tumor cell population, or the control of the area under the curve of the pharmacoki-
netics of a drug [31]. Here, we develop a novel mathematical proof of the summation laws,
now for a multitude of new variables reflecting the dynamics of biochemical and biologi-
cal networks. We base this proof on the concept of time-unit-invariance.
2. Results
2.1. The System
We will consider biochemical networks away from equilibrium in which various sub-
stances at well-defined concentrations (or mole numbers; we shall consider the volume of
the well-stirred compartment to be fixed) are connected by (mostly) enzyme-catalyzed
reactions. The latter may be chemical conversions or transmembrane transport processes.
Each reaction i between reactant molecules (‘substrates’) forms a number of products and
occurs at a certain reaction rate vi. The boundaries of the system of the networks are set
by fixed concentrations, fixed fluxes, or combinations of these. The concentrations and
reaction rates within the network are dependent variables that depend on and reflect the
state of the system. The latter is determined by all the fixed enzyme concentrations and
other parameters (we shall call independent variables ‘parameters’), such as the rate con-
stants of the enzymes, their kcat, their Michaelis and product inhibition constants, as well
as by the fixed external concentrations or fluxes. We assume that the time evolution of the
state variables is stable in the sense of Lyapunoff [32,33]. If its system parameters are left
unaltered, the system considered here will evolve into a stationary state, but the analysis
in this paper is not limited to such stationary states: it also addresses the time dependence
of concentrations and of other state variables, e.g., after a parameter has undergone an
Mathematics 2023, 11, 2473 3 of 11
instantaneous step change. In particular, we shall discuss the extent to which, at any point
in time, the state variables change magnitude when any of the enzyme activities has un-
dergone a permanent infinitesimal modulation at time zero. We shall prove a number of
laws that constrain the magnitudes of the corresponding [34] control coefficients.
2.2. Control Coefficients
A control coefficient quantifies the extent to which a catalytic process determines a
system variable. Originally control coefficients of metabolite concentrations and metabolic
fluxes were defined only for systems at steady state [3,4,34–36]. Their definition has since
been generalized to time dependent metabolite concentrations and fluxes [25,37] and to
properties characterizing time dependencies, such as cycling times, oscillation frequen-
cies, half times, and transit times [26,27]. Here, we write the definition of the control coef-
ficient of any dependent state variable x(t) in a biochemical network as:
𝐶
()≝𝜕ln (𝑥(𝑡))
𝜕ln (𝑒) (1)
𝑒 refers to the catalytic activity of enzyme j (or of any other process if it is not enzyme-
catalyzed); sometimes written as vj or as Vmax,j. This definition applies to any variable x,
which may be a concentration, a reaction rate, an electric potential, a time change of a
concentration, or the area under the curve (AUC) of an intracellular toxin concentration,
etc. All these together will populate the vector 𝑥(𝑡), which we here denote by x(t), with t
referring to time. We will here consider a set of, positive, x’s that can vary independently
of one another; pre-existing dependencies, such as through moiety conservation, must be
removed by transformations [28,30].
The vector 𝑒 (which will be denoted by e below) represents the complete set of cata-
lytic activities, each with a specificity j for a chemical or transport reaction, that can be
formulated explicitly and do not duplicate others: the e’s represent all parameters (inde-
pendent variables) with time dimensionality −1 (see below). For more complex systems,
for instance, with metabolite channeling, reaction steps involving multiple proteins may
need to be replaced by a vector of the corresponding rate constants, but we shall not deal
with such complications here [38,39]. The definition of the control coefficient should be
interpreted in the sense of an agent pj specifically affecting enzyme activity ej. The concen-
tration of that agent is altered instantaneously at time zero, and the effect on the variable
x(t) > 0 is determined. Therefore, more precisely:
𝐶()=⎝
⎛
𝜕ln(𝑥(𝑡))
𝜕𝑝
𝜕ln(𝑒)
𝜕𝑝⎠
⎞
. (2)
The time (t > 0) coefficient is defined as:
𝐶()=𝜕ln (𝑥(𝑡))
𝜕𝑙𝑛(𝑡) (3)
2.3. Setting the Time
The time coefficient may be rewritten (for t ≥ 0) as:
𝐶()=
()∙()
. (4)
This shows that for any actual development over time of the property x, ()
is well
defined. The time coefficient defined above is not yet uniquely determined, however. This
is because t depends on the choice made by the observer of when t should be called zero.
Choosing that time point at 50 rather than 2 min before the time point t would change the
Mathematics 2023, 11, 2473 4 of 11
time coefficient of x by a factor of 25 (through the factor t in the above equation). A similar
uncertainty exists for the property x. In order to remove these uncertainties, we consider
the type of system that we address more closely: deterministic Markovian systems. These are
systems of which the development over time after any given time point is unique and fully
determined by the magnitudes (initial values) of a number of state variables (called y) of that
system at that time point, plus two types of parameters. For these state variables y:
𝑑𝑦=
𝑓
(𝑘,𝑞,𝑦)∙𝑑𝑡. (5)
We shall call the magnitudes of time and state variables at the initial time point t0 and
y0, respectively (they may be set to zero later). In a metabolic system (biochemical net-
work) at a given temperature and pressure, the properties y are the concentrations of all
the molecules indicated by the vector y. One type of parameter is represented by the vector
k, which contains all parameters with dimension 1/time, including the rate constants and
the enzyme activities. The vector q represents all other parameters which do not have a
time dimension, such as equilibrium and Michaelis–Menten constants, standard chemical
potentials, and reaction stoichiometries. We assume that there are no parameters (inde-
pendent variables) with other time dimensionalities than −1 or 0. All these parameter val-
ues are considered to be fixed over the time span considered. When enzyme activities do
depend on tim e, e.g. , becau se of ge ne expressi on cha nges, this i s dealt with by Hierar chical
Control Analysis [40], or can remain part of the present analysis by describing them within
y. The above expression is a generalization of the time dependence of metabolite concen-
trations in metabolic networks, which is described by:
𝑑𝑦
𝑑𝑡=𝐍∙𝑣
=𝐍∙𝐸∙𝜑𝑦
,𝑘
, 𝐾
(6)
with φ a vector of enzyme rate laws, k a vector of rate constants and K a vector of Michae-
lis–Menten and equilibrium constants, E is a corresponding vector of enzyme concentra-
tions, N the matrix of reaction stoichiometries [30]. N and K are now subsumed in q and
the vectors will be further represented by the corresponding scalars. Again, we drop the
vector notation. Integration of the generalized equation then leads to:
𝑦(𝑡)=𝑦+
𝑓
(𝑘,𝑞,𝑦(𝑡))∙𝑑𝑡
. (7)
The state the deterministic system is in is fully defined by y, k and q and, thereby, by
y0, k, q, and t − t0. Consequently, all other state functions x are also determined by y0, k, q,
and t − t0.
Here, we will further focus on what we call ‘ideal biochemical networks’ [38]. Every
catalytic activity (including transport activities) in these is provided for by a specific pro-
tein (enzyme) and all these proteins function independently of each other. This means that
there is neither substrate channeling nor group transfer between proteins (when there are
such processes, the treatment becomes more complex but remains essentially the same).
In such networks, the set of rate constants k, can be replaced by the smaller set of enzyme
activities e. The initial concentrations can be added to the q parameters to constitute the pa-
rameter set p, all without time dimension. Consequently, the change since time zero of any
state variable x is a function of the enzyme activity vector e, the vector of parameters without
time dimension p, and the time t − t0:
𝑥−𝑥=𝑥(𝑒,𝑝,𝑡−𝑡). (8)
The parameter (independent variable) t (and t0 although not their difference) is am-
biguous as its value at any occurrence of interest depends on the moment in history at
which t is taken to equal zero. Multiplying dln(t) by
one obtains the fully defined
property
, which will not change when a different moment in time is taken as zero for
the time axis:
Mathematics 2023, 11, 2473 5 of 11
𝑑𝑙𝑛(𝜏)≝
∙𝑑𝑙𝑛(𝑡)=𝑑𝑙𝑛(𝑡−𝑡)=
. (9)
Consequently, one should substitute time parameter 𝜏≝𝑡−𝑡 for time t in the def-
inition of the time coefficient, or [25] take t0 = 0. This defines t0 as the time at which the
integration that was mentioned above starts and x0 as the corresponding ‘initial’ magni-
tude of x: 𝑥≝𝑥(𝑡). In summary, for the control coefficients to be unambiguously de-
fined one should either resort to the definitions:
𝐶
()=𝜕ln (𝜒(𝜏))
𝜕ln (𝑒) , (10)
𝐶
()=𝜕ln (𝜒(𝜏))
𝜕𝑙𝑛(𝜏) (11)
with: 𝜒(𝜏)≝𝑥−𝑥 (12)
and 𝜏≝𝑡−𝑡, (13)
or to the simpler definitions: 𝐶()=𝐶
()∙𝑥(𝑡)−𝑥
𝑥(𝑡) (14)
𝐶()=𝐶()∙𝑥(𝑡)−𝑥
𝑥(𝑡) ∙𝑡
𝑡−𝑡 (15)
with the proviso that t refers to the time elapsed since x equaled x0, and x refers to the value
of x minus x0. The simplest approach is then to set both x0 and t0 to zero as was conducted
by [25].
2.4. Summation Laws: Derivation
The observed magnitude of a system variable that does not have the dimension of
time should not depend on the unit that the observer uses to measure time. The observed
magnitude of a variable that does have a time dimension must depend on the time unit to
the extent that is precisely in accordance with that dimensionality. In order to illustrate
this, we consider two observers of the same natural phenomena. One observer measures
the time τ in hours and the other, referred to by τ′, measures it in minutes: τ′ is 60 times
larger than τ numerically, although the two times actually refer to the same moment:
𝜏=𝜆∙𝜏 (16)
with 𝜆 = 60 min/h in the example. System properties x may partially have a time dimen-
sion. Reaction rates, for instance, become 60 times smaller numerically when expressed in
moles per minute than in moles per hour, although physically they remain the same. More
in general for x′ expressed in the time unit of t′ and x expressed in time units of t:
𝑥′=𝜆∙𝑥 (17)
ρx represents the time dimensionality of x. For concentrations, thermodynamic efficiency,
growth yield (as flux ratio), electric potentials, cell transformation state, cell population,
and chemical potentials 𝜌=0. For reaction rates, transport rates, and fluxes 𝜌=−1. For
the half times, area under the curve (AUC), and mean residence time (MRT) much used
in pharmacology [31] 𝜌=+1. For ‘area under the moment curve’ (AUMC) 𝜌=+2.
Some parameters also have time dimensionality: enzyme activities (e) and rate constants
(k) have time dimensionality 𝜌=−1. The other parameters lack time dimensionality and
are represented by p.
Mathematics 2023, 11, 2473 6 of 11
The second observer will find: 𝑥′=
𝑓
(𝑒′,𝑝′,𝑡′−𝑡′). (18)
This will compare as follows with the observations made by the first observer:
𝜆∙x𝑡,𝑒,𝑝=𝑥′=x(𝜆∙𝜏,𝑒/𝜆,𝑝). (19)
Since this should be true for any value of λ > 0, we can equate the logarithmic deriv-
ative with respect to ln(λ) of the left-hand side of Equation (19) to the same derivative of
the right-hand side of that same equation. After rearranging the equation, we find:
𝜌=()
(∙)∙(∙)
()+∑()
∙
()
+∑()
()∙()
()
=()
(∙)−∑()
. (20)
For 𝜆=1, and after multiplication by
(if 𝑥≠0 unless 𝑥=0; see above) this
becomes the generalized summation law for time dependent control coefficients 𝐶 (we
write x for x(τ)): 𝜌∙
+∑𝐶
=𝐶. (21)
For concentrations y, this implies that at any time point [25]:
∑𝐶
=𝐶, the sum being taken over all catalytic activities ej. (22)
The same should be true for the control of thermodynamic efficiency, the control of
flux ratios and concentration ratios, the control of electric and chemical potentials, etcet-
era. The sum over all enzymes (catalytic activities) in the network of the concentration
control coefficients of any substance should herewith be positive when its concentration
is on the increase with time, negative when it is on the decrease. When it is at its maximum
or just steady that sum should be zero:
∑𝐶[]
=𝐶[]=,, 0. (23)
The formulation for steady state (Equation (23)) is the classical summation law for
concentration control coefficients [3,4]. The same law applies to control of transmembrane
electric potentials, phosphorylation potentials, and DNA supercoiling. For any reaction
rate v at time point τ: ∑𝐶
=𝐶+
. (24)
Assuming (see above) that at t = t0 = 0 the system was started from a state at zero flux
so that v0 = 0, this becomes: ∑𝐶
=𝐶+1. (25)
This implies that when the flux is at a maximum (or minimum), or the system is at
steady state, increasing all activities in proportion will increase the flux in the same pro-
portion, because then 𝐶=0. Only when the flux is decreasing with time at a time coef-
ficient of −1, may such a collective increase in process activities leave the flux unaffected.
The steady state case (Equation (25) at 𝐶=0) is the classical flux-control summation law
[3,4].
Let the “area under the curve up to time point t” (AUCt; [31]) be the time integral of
the variable concentration of a substance in the compartment of interest:
𝐴
𝑈𝐶𝑡≝𝑦∙𝑑𝑡
. (26)
As this has the dimension of time, the summation law predicts:
∑𝐶
=𝐶−1. (27)
Mathematics 2023, 11, 2473 7 of 11
When all xenobiotic has left the body, time no longer affects the AUCt so that
𝐶 =0. AUCt then becomes the AUC known in pharmacology, for which the sum of
the control coefficients should equal −1:
∑𝐶
=−1. (28)
The “area under the first moment curve up to time point t” is defined by [31]:
𝐴
𝑈𝑀𝐶𝑡≝𝑦∙𝑡∙𝑑𝑡
(29)
and has a time dimensionality of +2. Accordingly, its summation law reads:
∑𝐶
=𝐶−2. (30)
As the xenobiotic leaves the system, the AUMCt also becomes constant in time and
the summation law reads: ∑𝐶
=−2. (31)
The mean residence time up to time point t is defined by [31]:
𝑀𝑅𝑇𝑡≝𝑦∙𝑡∙𝑑𝑡
𝑦∙𝑑𝑡
(32)
and, thereby, has a time dimensionality of 1. Consequently:
∑𝐶
=𝐶−1. (33)
When all xenobiotic has left the system, the MRTt becomes the mean residence time
MRT, and the total control exercised by the enzymes equals −1.
Considering the concentration-versus-time curve of a xenobiotic after its injection
into the body, one may wonder at what time a certain concentration is reached. One then
sees the time as a function of that concentration (and of all the enzyme activities). We
consider a modulation of all enzyme activities by the same factor, so that for all i’s 𝑑𝑙𝑛𝑒=
𝑑𝑙𝑛𝑒. We allow for a simultaneous modulation of time to such an extent that there is no
change in y. This leads us to:
0=𝜕ln(𝑦)
𝜕ln(𝜏)∙𝑑𝑙𝑛𝜏
𝑑𝑙𝑛(𝑒) +𝜕𝑙𝑛(𝑦)
𝜕ln(𝑒)∙𝑑𝑙𝑛(𝑒)
𝑑𝑙𝑛(𝑒)
=
=𝜕ln(𝑦)
𝜕ln(𝜏)∙𝐶
+1
(34)
where we have used the corresponding summation law as well as the expression:
𝑑𝑙𝑛(𝜏)=∑
∙𝑑𝑙𝑛𝑒
(35)
and the definition of the control coefficients of the time at which the curve reaches y, i.e.,:
𝐶≝
. (36)
The general solution is that the sum of the control coefficients of the specific time
point (𝜏) at which the curve reaches the concentration y equals −1:
∑𝐶=−1
. (37)
This will be true for any concentration of the substance, and, therefore, also for where
it is half the maximum value, either on the increasing or on the decreasing slope, and
where it is at the maximum.
Mathematics 2023, 11, 2473 8 of 11
3. Discussion
We have here derived a number of summation laws constraining the control of prop-
erties of metabolic and other networks in Biology. For control with respect to concentra-
tions, these summation laws had been derived before by using different methodologies.
[25] used the phenomenon that an acceleration of all processes by a factor should make
everything happen in the same way but at a time point earlier by that factor. Our approach
to changing the time unit in which the system is observed is similar to this. For the steady
state summation laws, others and ourselves have used the property that steady state
fluxes and concentrations are homogeneous functions of enzyme activities [29,30], per-
formed the corresponding thought experiments [3], or have taken derivatives of the bal-
ance equation at steady state [4,28].
In the present paper, we have found summation laws for many more properties than
the previous works had found, such as for read-outs of pharmacokinetics (AUMC), results
of microbial growth experiments (yields), non-equilibrium thermodynamic properties (ef-
ficiencies), chemical potentials and Gibbs energy differences of reactions, and half times
of dynamic changes in concentrations. In fact, the summation laws we proved here are
valid for state variables of any time dimensionality ρ.
The summation laws have multiple implications for the control of dynamic phenom-
ena [41]. We here mention only a few examples: When a concentration is at its time max-
imum, the corresponding summation law implies that this maximum concentration can-
not just be determined by a single reaction activity in the system; there must be at least
one additional controlling activity with an opposite sign. This is important for oncology,
as it proves that the control over the phosphorylation state of an important ‘onco-protein’
must be shared by at least two other gene products. The steady state thermodynamic ef-
ficiency of microbial growth cannot be determined by a single process activity either. This
is important because energy processes and microbial growth may be optimal in part in
terms of growth rate, in part in terms of growth yield, and in part in terms of the thermo-
dynamic efficiency of growth [30,42]. For the half times, the summation law implies that
if one activity controls that time, then there must at least be one other activity in control
unless a factor increase in the former causes a reduction in the half time by the same factor.
This is important for the understanding of the control of dynamic signaling by the MAP
kinase cascade [43]. The sums for the area under the curve (−1) and for the ‘area under the
first moment curve up to time point t’ (−2) are again new findings and of great potential
interest to pharmacology, where these variables are used to characterize the pharmacoki-
netics of clinical drugs [31]. When some enzyme activity is limiting for a biotechnological
or medical process, one often tries to activate it. This will then reduce its control coeffi-
cient. The summation law has the implication that this automatically makes some other
process more limiting, suggesting a second candidate for optimization [44].
Of course, the existence of the summation laws depends on the way ‘control’ is de-
fined. It hinges on taking the double logarithmic derivative, which corresponds to the
percentage increase of the controlled variable resulting from a 1% activation of a process
[30,34]. It also depends on limiting the controlling factors to the set of catalytic activities:
the control coefficients are but a subset of all possible sensitivity coefficients. Examining
systems in terms of more sensitivity coefficients than the control coefficients can lead to
more insight into the why’s and how’s of their design and functioning [45–48], but not to
these summation laws. Formulation of the control in terms of straight rather than log–log
derivatives [28] changes the summation laws to equations that are so complex that their
meaning may elude the biologically interested reader.
For the summations to lead to the fixed numbers found here, i.e., for the properties
to become laws, the set of control coefficients (over which the sum is taken) should be
complete. This completeness means that all parameters with time dimension should be
subsumed in the set, either directly or indirectly, because their effects can be represented
by a modulation of the enzyme activities [49]. The completeness is served by the advent
Mathematics 2023, 11, 2473 9 of 11
of both genomics and systems biology with their interest in producing the complete ge-
nome and proteome of organisms [50] and with the vision of making genome-wide meta-
bolic maps of a variety of organisms [1,51,52].
It is occasionally suggested that summation laws pertain to sums over all enzymes in
the pathway of the flux or concentration under consideration. The above derivation shows
that this is not so: the sum is over all the reaction activities in the entire network. Indeed,
steps with major control may reside outside the pathway proper [53]. That the control may
be distributed over the entire genome-wide network explains why so many genes exert so
little control in biology, notwithstanding the ubiquitous myth that every biological function is
determined by a single ‘key’ gene or enzyme. Where Kacser and Burns noted this for the con-
trol of fluxes [2,54], we may now generalize to all functional properties of complex Life, if
not complex society [55].
Our results are important for Biology at large as they prove that general principles,
such as invariance to unit transformation [56], are not confined to Physics. Many of these
principles extend to Life sciences, with consequences that are illuminating and important
for Biology and with instantiations that are less important for Physics and inorganic
Chemistry. An example is the issue of whether the first irreversible step in a pathway is
the rate-limiting step, with all subsequent steps being irrelevant for the control of pathway
flux. The summation law states that there should indeed be a total rate limitation (i.e., the
sum of flux control coefficients) of one but that this may be distributed over the pathway
steps. For the simpler pathways in Physics and Chemistry, the distribution is such that all
control is in the first irreversible step. In Biochemistry, control tends to reside in demand
rather than supply [57], probably as a result of evolutionary selection for the organism’s
fitness through responsiveness to changes in workload. The way evolution has achieved
this is by developing product inhibition of the enzymes [30], a phenomenon that is absent
from inorganic Chemistry and Physics since they lack enzymes. More specifically, the pre-
sent paper is important because it generalizes these summation laws to many properties
(dependent variables) of biological interest, including cell cycle time, specific growth rate,
growth yield, growth efficiency, transformation state, elasticity (response) towards a me-
tabolite or signaling molecule, survival probability, DNA structure, and epigenetic state:
as proven above, the numbers to which the corresponding control coefficients sum are
given by the time dimensionality of the variables.
Funding: This research received no external funding.
Data Availability Statement: All material that could be shared is in this paper.
Acknowledgments: This paper is dedicated to the late Reinhart Heinrich and the late Henrik Kacser
who have done so much for the understanding of biology and life, with their Metabolic Control
Analysis that turns 50 this year.
Conflicts of Interest: The author declares no conflict of interest.
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