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Evolving Biological Systems: Evolutionary Pressure to Inefficiency

01 Jan 2014-

TL;DR: It is found that competition can lead to an evolutionary pressure towards inefficiency, that is less growth is achieved per unit of input nutrient in a model of nutrient uptake.
Abstract: The evolution of quantitative details (i.e. “parameter values”) of biological systems is highly under-researched. We use evolutionary algorithms to co-evolve parameters for a generic but biologically plausible topological differential equation model of nutrient uptake. In our model, evolving cells compete for a finite pool of nutrient resources. From our investigations it emerges that the choice of values is very important for the properties of the biological system. Our analysis also shows that clonal populations that are not subject to competition from other species best grow at a very slow rate. However, if there is co-evolutionary pressure, that is, if a population of clones has to compete with other cells, then the fast growth is essential, so as not to leave resources to the competitor. We find that this strategy, while favoured evolutionarily, is inef- ficient from an energetic point of view, that is less growth is achieved per unit of input nutrient. We conclude, that competition can lead to an evolutionary pressure towards inefficiency.
Topics: Evolutionary pressure (53%), Population (53%)

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Citation for published version
Chu, Dominique and Barnes, David J. (2014) Evolving Biological Systems: Evolutionary Pressure
to Inefficiency. In: ALIFE 14: The Fourteenth Conference on the Synthesis and Simulation
of Living Systems. MIT Press pp. 89-96.
DOI
https://doi.org/10.7551/978-0-262-32621-6-ch016
Link to record in KAR
https://kar.kent.ac.uk/44781/
Document Version
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Evolving biological systems: Evolutionary Pressure to Inefficiency
Dominique Chu
1
and David Barnes
1
1
School of Computing, University of Kent, CT2 7NF, Canterbury
dfc@kent.ac.uk
Abstract
The evolution of quantitative details (i.e. “parameter values”)
of biological systems is highly under-researched. We use evo-
lutionary algorithms to co-evolve parameters for a generic but
biologically plausible topological differential equation model
of nutrient uptake. In our model, evolving cells compete for
a finite pool of nutrient resources. From our investigations
it emerges that the choice of values is very important for the
properties of the biological system. Our analysis also shows
that clonal populations that are not subject to competition
from other species best grow at a very slow rate. However,
if there is co-evolutionary pressure, that is, if a population of
clones has to compete with other cells, then the fast growth is
essential, so as not to leave resources to the competitor. We
find that this strategy, while favoured evolutionarily, is inef-
ficient from an energetic point of view, that is less growth is
achieved per unit of input nutrient. We conclude, that com-
petition can lead to an evolutionary pressure towards ineffi-
ciency.
Introduction
Much is now known about biological systems at the molec-
ular level. T here are cou ntless databases that contain giga-
bytes of de ta iled information abo ut biochemical networks,
reactions, gene regulation, p rotein-protein interac tions and
much more. As far as biochemical reaction networks are
concerned, most of the available information is abo ut struc-
tural properties of these networks, i.e. which molecules re-
act with w hich molecule, which p rotein represses/activates
which gene and so on. At the same time, very little is known
about the quantitative details of these re actions, i.e . how fast
reactions proceed , how strong a gene is repressed or at what
rate genes are expressed.
Recently, a large scale analysis of topological data has
led to important insights into the design principles of living
systems. The discovery of so-c alled network motifs(Alon,
2007; Kashtan and Alon, 2005; Mangan and Alon, 2003),
i.e. over-represented local connectivity patterns of gene reg-
ulatory networks is but one example. These motifs were
found to be not on ly statistically over-represented but also
functionally significant(Alon, 2006). Wh ile mu ch research
effort has been expended to understand the significance of
these top ological features, very little research has been done
to understand quantitative details of biochemical reaction
networks(Chu, 2 013).
One of th e conditions for being able to gain insight into
the topological design principles of biological systems was
the wealth of empirical available about them. Since there is
not a comparable amount of information available about the
parameter values of biochemical networks, it is only natural
that much less is known ab out the quantitative de sig n prin-
ciples of natural systems. At the same tim e , it is likely that
the values of param eters of biological sy stems conta in very
much biologica lly valuable information . They are a product
of natural evolution and as such have to be assume d to reflect
the adaptive pressures to which th e system has been exposed
and as such encode valuable biological in formation.
In order to u nderstand the principles that guide the evo-
lution of q uantitative parame ter values, it is not nece ssary
to know the actual values of biological systems. Instead, a
different approach based on synthetic evolution using evolu-
tionary algorithms can be used. In this article we will take
this approach. To do this we focus on a generic topologi-
cal model o f nutrient uptake, i.e. a model that only contains
the structure of the biochemical r e actions, but not their nu-
merical parameters. We then use evolutionar y algorithms to
evolve parameters for specific conditions. Comp a risons of a
large number of runs will the n enable us to draw some con-
clusions as to how pa rameters evolve. The hope is that th ese
conclusions are valid beyond the specifics of the particular
model we have c hosen and provide insigh t about na tural bi-
ological systems a s well.
Our model does not describe any specific biological sys-
tem, but it is a biolog ically plausible generic representa-
tion of n utrient uptake in bacteria and contains topologi-
cal fe a tures that are widely used b y bacteria. An important
way for bacteria to take up nutrient is by importing nutrient
molecules through specialised openings at the cell surface—
so ca lled porins. These porins are proteins and they tend to
be specific to a particular nutrient type. So , a porin for one
nutrient cannot be used to take u p a different type of nutrient.
In bacteria, these porins whose production requires energy
ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems
DOI: http://dx.doi.org/10.7551/978-0-262-32621-6-ch016

are only expressed by the cell when the relevant nu trient is
actually present in the environment. A typ ic a l way for the
cell to achieve this is to use the nutrient as an activator for
the expression of th e porin. Once imported into the cell the
nutrient stimulates the expression of the gene codin g for the
porin (indeed, often it represses the repression of the gene,
which amounts to stimulation). This very g eneral scheme of
porin activation is reflecte d in our model.
A typical feature of bacterial uptake system is that expres-
sion is demand driven and porins are on ly prod uced when
they are needed. The evolutionary ratio nale for this is that
gene expression requires resources that could be invested
otherwise, for example to fu el growth. Moreover, there is
finite space on the cell surface which limits the number of
porins that can be expressed at any one time. It is also com-
monly observed that over- or under-expressing a gene of te n
decreases the growth of the mutant strains. So, a pparently,
for many proteins there is an optimal rate of porin exp res-
sion. At the same time, evolution has the ability to tune
the rate of some biochemical reactions, including the rate of
gene expression. I t is therefore likely that the pa rticular rates
of gene expression and that of other bio-chemical reactions
are fine-tune d by evolution.
This motivates the r esearch question to be addressed in
this contribution: How do the par ameters of generic bacte-
rial uptake systems dep e nd on the adaptive p ressures that led
to their emergence. Moreover, given a set of adaptive pre s-
sures, is it possible to predict the parameters, or vice versa,
given a set of parameters, is it possible to un derstand what
adaptive pressures led to them? Finally, can the resu lts ob-
tained from the generic biologically plausible model provide
any insight that is relevant for the real world.
To add ress these questions, we performed two types of
artificial evolution experiments. Firstly, we evolved parame-
ters (i.e. “solutions”) on their own. We foun d that this results
in uptake mechanisms that could turn most of the nutrient on
offer into growth using a very low number of porins result-
ing in slow nutrient uptake and g rowth. We found this to
be the most efficient mode of g rowth because it allows the
cell to channel mo st nutrient into growth while minimising
the amount of energy spent on the uptake mechanisms. In a
further set of experiments we then evolved new solutions in
competition with a previously evolved one.
The solutions obtained from these co-evolutions were
different from the solutions evolved withou t competition.
Rather than taking up n utrients slowly with a low number
of porins, they evolved towards increasingly rapid uptake
of nutrient (a lthough not necessarily rapid growth). While
this allowed them to grow fast it also means, as we will d is-
cuss below, that they grow inefficiently. Sp ecifically, we
found a clear trend that co-evolved solutions are less effi-
cient than the or iginal solutions that evolved without a com-
petitor. However, within the chain of evolved solutions there
was no clear further trend toward inefficiency. Hence, rather
Figure 1: A schematic re presentation o f the model.
than getting more efficient by competition, we found that co-
evolution leads to less efficient solutions, which is a co unter-
intuitive at first. We will argue below that this pattern to-
wards inefficiency is universal, in the sense that it doe s not
depend on the specifics of the particular model, but would
be true for a large cla ss of nutrient uptake systems, includ-
ing th ose of real organisms.
Furthermore, in ou r simulatio ns we presented the simu-
lated c ells with two different types of nutrients of differing
quality. We also added a structural m otif into the model that
would allow the cells to supp ress take-up of the less valu-
able nutrient 2 in favour of the other. Indeed, we observed
the evolution of the suppression of nutr ient 2 uptake. How-
ever, sur prisingly to us, the solutions did not use the motif
offered, but instead came up with a different way of regulat-
ing th e uptake of the less efficient nu trient.
The basic model
We present here a generic model of a bacter ia l up-
take/metabolic system (see figure 1). The idea is that there
are two sources of nutrients N
1
and N
2
. Uptake of these
sources of nutrients requires specific porins, namely P
1
and
P
2
respectively. Once taken up into the cell the nutr ient be-
comes an internal source of energy ( E
1
and E
2
) which can
be converted into actual energy (or ATP), which we deno te
by E
0
. We a ssume that th e uptake and conversion of nutrient
follows Hill kinetics(Chu et al., 2011). Th e internal energy
is converted either into porin s (i.e. porin 1 and porin 2 ab-
breviated as P
1
, P
2
) or into bio mass (bm) which represents
the results of bacterial growth.
We only dete rmined the topology of this model which is
designed such that the expression of p orin 1 and 2 is acti-
vated by the presen ce of nutrient 1 and 2 within the cell (i.e.
E
1
and E
2
respectively). The model topology does no t by
itself specify how strong this activation is. The strength of
the a c tivation depends on the parametrisation, which need s
to be evolved. Indeed, there are many parameters that would
effectively turn off the activation. The same is true for all
other regulatory functions in the model.
An important feature of the model is th a t the exp ression
of nutrient and the production of biomass requ ire energy.
ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems

Hence, the (a priori unspecified) parameter values for the
expression rates of porins and the growth ra te decide to what
extent the resources (i.e. nutrient) is used to fuel growth
and to what extent it is used to maintain th e cellular uptake
machine, i.e. how much is allocated to porin p roduction.
It appears that there is an optimal allocation of re source
to growth and the uptake mechanism. If the cell allocates
no energy to uptake but all to growth, it will not be able to
use any of the nutrients and hence it will not grow at all. On
the other hand, if the cell allocates all of its nutrients into
uptake, but none into growth, then it will be rich in nutri-
ents, but never grow and hence never divide. In-between
these two extremes there is one (or possibly several) optimal
allocation. While it is clear fro m this argument that such
an optimum exists, we do not kn ow where it is and what it
depends on.
Another important fe a ture of the model is that the total
number of porins in the system is limited. Porins in bac teria
are loca ted at the cell surface. They build op enings there
and selectively let molecules in and out of the cell. In real
cells there is limited sp a ce on the surface to accommodate
porins. This limitation is represented in our model by th e
term L (see below). It is a repressing term that reduces the
expression of p orins 1 a nd 2 as a (Hill-repressor) function o f
the sum of the concentration of both.
Finally, the model also features a repressor motif. The
molecule R is expressed when there is porin 1 available in
the cell and its sole purpose in the model is to repress the
expression of porin 2. This sort of regulatory motif whereb y
a repressor is activated by some part of the system and re-
presses another part o f the system is commonly found in
gene regulatory networks. Th e idea of introducing this mo-
tif is to enable the cell to evolve a repression mechanism for
nutrient 2 when the (better) nutrient 1 is available.
The topology of the model can be summarised by these
chemical equations:
N
i
ǫ
P
i
E
i
, k
N
i
p
i
N
i
N
i
+ K
N
i
E
i
E
0
, k
E
i
E
i
P
1
+ E
0
P
1
, k
P
1
E
h
P
1
1
E
h
P
1
1
+ K
h
P
1
P
1
E
0
L
P
2
+ E
0
P
2
, k
P
2
E
h
P
2
2
E
h
P
2
2
+ K
h
P
2
P
2
K
h
R
R
R
h
R
+ K
h
R
R
E
0
L
P
0
k
P
0
P
i
P
i
+ K
P
i
r + E
0
R, k
R
E
0
E
1
E
1
+ K
R
{P
i
, E
i
, R} , d
{P
i
,E
i
}
E
0
bm, k
C
(1)
where L is the space-limit which represents the fact that
there is limited space at the surface of cells to accommodate
porins, given by
K
L
(P
1
+ P
2
) + K
L
The quality factor ǫ
P
i
determines the quality of a nutrient
and we set it to 1 for P
1
and 0.5 fo r P
2
. This m eans that one
unit of nutrient 2 gives only 1/2 unit of biomass. Up take and
gene expression are assumed to follow H ill kine tics. While
this is an approximation, in reality it has been found that
Hill k inetics is a good description of the reactions de scribed
here. It is also widely used to model them and is a fairly
simple approach. In all simu la tions reported here we keep
the Hill exponent fixed at a value of 2, which is biologically
plausible.
Evolving the sys tem
In this article we evolve parameters for the topological
model described by equ a tion 1. Concretely this m eans that
we evolve values fo r the kinetic parameters determining the
system, including the Hill-constants (i.e. K
N
i
and dynamic
constants such as k
P
i
. No te that we do not evolve the decay
rate d which we keep fixed at 0.1, the Hill exponents (i.e.
h
x
= 2), the relative value of ǫ
P
i
(which we keep fixed at 1
and 0.5 respectively) and K
L
which determines how much
space there is for the porins in the cell. This latter para meter
we set to 1. All other parameters are evolved and we allow
them to take values between 0 and 15. In all simulations
reported here the mod el is implemented as a system of dif-
ferential equations. As a solver we use the general purpose
numeric differential equation solver of the Maple computer
algebra system version 16 for Linux.
The model was implem ented as a co-evolutionary system,
that is we have two different solutions compete for the same
nutrient pool o f N
i
. T his represents two different species
of ba c te ria co-existing in the same environm ent. In practice
this means that we used two sets of differential equations
with two sets of the variables E
i
, P
i
, R, bm representing two
different cell-types. Each set had the ir own kinetic para m-
eters, yet the ir dy namics depended on one another v ia the
shared nutrient pool. Of the two competing solutions, we
ever only evolved one of those solutions, while keeping the
other one fixed. Initially, we use as the fixed solution an
“unfit standard so lution” with all parameters set to 1. This
solution supports no growth beyond the start-up allocation
which is equivalent to 1 unit of biomass. Co-evolution is
achieved by using previously evolved solutions as fixed so-
lutions (i.e. “incum bents”) in further evolutionary r uns. In
all simulations we set as the initial cond ition all variables to
zero except for P
i
= 0.001, E
0
= 1. This means that any
solution ca n support a maximum of 1 unit of biomass even
if it does not take up a single unit of nutrient.
During each evolutionary run only one of th e solutions is
evolved, wh ile the other one is kept fixed at user-defined pa-
ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems

rameters. Co-evolution was achieved in a sequential man-
ner, that is, one solution was evolved against a fixed so-
lution. The evolved solution was then subsequently used
as the fixed incumbe nt in a further evolutionary run . Co-
evolutionary chains were obtained as follows;
1. Evolve a first so lution again st an un-evolved ba se solution
(all parameters set to 1).
2. Once the first solution is obtained, evolve a second solu-
tion against the first solution (which is kept fixed).
3. Create a third solution by evolving against the second so-
lution (which is now also kept fixed).
4. Continue in this m anner until no more solutions evolve.
To evolve the system we used a genetic algorithm with
elitism. Individual solutions were represented as an array of
real numbers in the ran ge [0, 15]. The population size was
set to 50. The initial population consisted of random param-
eters w ithin the range [0, 15] sampled from a uniform distri-
bution. As a fitness function we chose the biomass after 500
units of time. We found that 500 time units was large com-
pared to the transient periods of the system, i.e. increasing
this time did not ch ange th e r esults of the evolution.
As a selection algorithm we c hose a fitness proportional
selection. However, in every generation the best solution
and a mutated version of it was a llowed to proceed to the
next population. The mutation and crossover rate was set to
0.8. Mutation was done by changing a random parameter by
up to ten per cent of its current value. If a mutation resulted
in a value lower than 0.00001 or grea te r than 15 then the pa-
rameter was set to 0.00001 and 15 respectively. The amount
of available nutrient was set to 10 for both nutrient types.
The G A was implemented in Perl, but the fitness function
was evaluated using Maple . Both the relevant Maple script
and the Perl source code are available from the authors upon
request.
We performed two different types of experim ents. Firstly,
we performed a simple evolution without competition (i.e.
with the standard unit solution as compe titor). Subsequently,
we used the results of those evolutionary simulations to initi-
ate a co- evolutionary chain, as described above. In practice
we found that after a number of iterations no more fit so-
lutions were fo und, in the sense tha t the total b iomass pro-
duced for the evolving solution did not substantially exceed
1, i.e. evolution could not find solutions to outperform the
incumbent. In this situation it was helpful to evolve a new
solution by seeding the new evolutionary solution with the
incumbent parameters, ra ther than starting from a random
solution. However, even in this case, the co-evolutionary
potential was limited.
Individual evolutionary r uns we re stopped either after
5000 generations or when a plateau of high fitness with no
apparent further increases over time was reached, whatever
happened first. The presence of such plateaus was deter-
mined by visual inspection. I n practice, it turned out to be
a clear-cut case. A ty pical evolution would show rapid in-
creases of the fitness at first, followed by fitness stagnation.
Results
Unconstrained evolution
We evolved a number of solutions without competitor. Fig-
ure 2 illustrates three typical results obtained from unco n-
strained evolution. It shows the amount of biomass over time
obtained by simulating in Maple the best so lution of the fi-
nal population in the GA. It is part of the set-up that there
is a limited amount of nutrien t of 20 units divided ac ross
two type s of nutrients. Since the second nutrient gives only
half the growth of the first, at best the available resource can
be converted into a biomass of 15 units under ideal condi-
tions; the solutions also get a start-up energy equivalent to 1
biomass. Hence, in total the maximum they can reach is 1 6
biomass u nits.
It is ap parent from figu re 2 that most solutions evolved
come c lose to the maximum attainable biomass, although
there is so me variation. Occasionally, we have also ob served
that solutions got stuck on a local minimum and did not dis-
cover the second nutrient so urce. This resulted in cells that
would not take up any of the nutrient 2 and achieve only a
level of abou t 10 units of biomass (data not shown). This in-
dicates that the solu tions were able to channel most nutrients
into growth rather than using the m for enzyme pro duction.
This high level of co nversion was made possible by a very
low assumed degradation rate of en zymes that allowed the
solutions to grow at a slow rate.
The figure shows that the time required for achieving the
maximal growth varies somewhat from solution to so lution.
The three example solution shown in figure 2 are representa-
tive for the range obser ved in all unconstraine d evolutionary
runs, Generally, we observed that these evolved solutions
take up nutrient over a time period of 20 to 150 time units.
There is a wide variation between the solutions that we ob-
tained.
Co-evolution
Co-evolution chan ges the nature of the solu tion obtained in
very spec ific ways. The system as a whole offers a finite
amount of resources and both solutions need to c ompete for
the same two pots o f nutrients. Hence, competition is not a
zero su m game.
At the b egin ning of a co-evolutionary run the co mpeti-
tor will have random parameters and not be able to compete
well against the incumbent. However, as new solu tions are
discovered the competitor evolves to outp erform the in c um-
bent. One way to do this is to consume the available nu -
trients faster than the incumbent. Idea lly, the new solution
has used up all of the nutrients before the incumben t can
ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems

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TL;DR: Network motifs are reviewed, suggesting that they serve as basic building blocks of transcription networks, including signalling and neuronal networks, in diverse organisms from bacteria to humans.
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S. Mangan1, Uri AlonInstitutions (1)
TL;DR: This study defines the function of one of the most significant recurring circuit elements in transcription networks, the feed-forward loop (FFL), which is a three-gene pattern composed of two input transcription factors, both jointly regulating a target gene.
Abstract: Engineered systems are often built of recurring circuit modules that carry out key functions. Transcription networks that regulate the responses of living cells were recently found to obey similar principles: they contain several biochemical wiring patterns, termed network motifs, which recur throughout the network. One of these motifs is the feed-forward loop (FFL). The FFL, a three-gene pattern, is composed of two input transcription factors, one of which regulates the other, both jointly regulating a target gene. The FFL has eight possible structural types, because each of the three interactions in the FFL can be activating or repressing. Here, we theoretically analyze the functions of these eight structural types. We find that four of the FFL types, termed incoherent FFLs, act as sign-sensitive accelerators: they speed up the response time of the target gene expression following stimulus steps in one direction (e.g., off to on) but not in the other direction (on to off). The other four types, coherent FFLs, act as sign-sensitive delays. We find that some FFL types appear in transcription network databases much more frequently than others. In some cases, the rare FFL types have reduced functionality (responding to only one of their two input stimuli), which may partially explain why they are selected against. Additional features, such as pulse generation and cooperativity, are discussed. This study defines the function of one of the most significant recurring circuit elements in transcription networks.

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"Evolving Biological Systems: Evolut..." refers background in this paper

  • ...The discovery of so-called network motifs(Alon, 2007; Kashtan and Alon, 2005; Mangan and Alon, 2003), i.e. over-represented local connectivity patterns of gene regulatory networks is but one example....

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