A case study of evolutionary computation of biochemical adaptation

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IOP PUBLISHING PHYSICAL BIOLOGY

Phys. Biol. 5 (2008) 026009 (12pp) doi:10.1088/1478-3975/5/2/026009

A case study of evolutionary computationof biochemical adaptationPaul Francois and Eric D Siggia

Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Avenue,10065 New York, NY, USA

Received 11 April 2008Accepted for publication 2 June 2008Published 24 June 2008Online at stacks.iop.org/PhysBio/5/026009

AbstractSimulations of evolution have a long history, but their relation to biology is questioned becauseof the perceived contingency of evolution. Here we provide an example of a biologicalprocess, adaptation, where simulations are argued to approach closer to biology. Adaptation isa common feature of sensory systems, and a plausible component of other biochemicalnetworks because it rescales upstream signals to facilitate downstream processing. We createrandom gene networks numerically, by linking genes with interactions that modeltranscription, phosphorylation and protein–protein association. We define a fitness functionfor adaptation in terms of two functional metrics, and show that any reasonable combination ofthem will yield the same adaptive networks after repeated rounds of mutation and selection.Convergence to these networks is driven by positive selection and thus fast. There is always apath in parameter space of continuously improving fitness that leads to perfect adaptation,implying that the actual mutation rates we use in the simulation do not bias the results. Ourresults imply a kinetic view of evolution, i.e., it favors gene networks that can be learnedquickly from the random examples supplied by mutation. This formulation allows fordeductive predictions of the networks realized in nature.

1. Introduction

Evolution is a retrospective theory, it explains the similaritiesamong genes or organisms as a consequence of commonancestry. However, if one wants to predict the sequence ofa protein that has a desired activity, or a gene network thatrealizes some function, Darwinian evolution does not providean answer. The impediments to true prediction in biologyare (i) lack of a mathematical expression for the fitness, thequantity to be optimized, (ii) ignorance of mutation rates and(iii) inability to infer phenotype from genotype. In this paperwe will show how these difficulties can be overcome for theproblem of adaptation.

The choice of fitness (more accurately termed a fitnessfunction since it assigns a number to an arbitrary geneticnetwork) is analogous to setting up a genetic screen. Itdefines the problem in quantitative terms. Our ignorance ofthe mutation rates is circumvented by showing for the chosenfitness function that there is a smooth monotone path leadingto at least a good local optimum in the fitness. So like a rocktumbling down a valley, the bottom is unique even though thepath leading to it may have arbitrary jigs and jags.

1.1. Defining a proper fitness function

Genetic algorithms have become a standard component in thetoolbox of techniques for nonlinear optimization of a costfunction. They mimic biology by following a ‘population’ ofpotential solutions that are ‘mutated’ by changing each one intoa related solution in a random way. The most ‘fit’ solutions,as measured by the cost function, are replicated and replacethe less fit ones. The topography of the cost function and theway mutations sample it control the convergence rate to anoptimum. If the topography is a funnel leading to a uniqueminimum, convergence for any reasonable mutation processis assured. For a ‘golf course’ with multiple local minima andno cues about proximity to any one of them, there will be longperiods of neutral evolution and occasional trapping.

Both the fitness function and the mutation process arehard to quantify in biology. For simulations, we need aprecise fitness function that can be evaluated for any systemeven if far from optimum. The stock answer of reproductivesuccess is not helpful since it depends on too many unknownfactors, when in fact all we can hope to evolve is somesmall part of an organism such as a signaling pathway. Our

1478-3975/08/026009+12$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

Phys. Biol. 5 (2008) 026009 P Francois and E D Siggia

definition of fitness should be guided by a salient fact of organicevolution that pathways do not evolve in isolation, but areshared among many species and reused in many contexts. Thefitness function should select for the generic features of thesystem, and ideally provide cues that will direct an arbitrarygenetic network along a path of continuous increasing fitness.This both makes for efficient numerical search but also hasimplications for how biological evolution may operate.

In some cases there is a natural fitness function, such asvisual acuity in the case of the eye [1], from which the authorsdemonstrated a monotonic adaptive path from a photosensitivepatch to a compound eye. Another plausible fitness functionwas segment number for somitogenesis [2]. Interesting clockand bistable systems were obtained in [3] and particularnetworks for adaptation in [4], but in both cases the choiceof fitness function was ad hoc.

More typical is the situation where multiple criterionimpinge on fitness and the tradeoffs among them are notapparent and likely organism or context dependent. It isnot a matter of imposing additional numerical criterion on ageometric quantity such as segment number (e.g., demandingsegments of a particular spacing or size). If biological clocksevolved to anticipate light–dark cycles, how should one weighthe persistence of the anticipation, the strength of the response,its sensitivity to the phase of the light source and not itsamplitude, etc. To only reward the perfect circadian oscillatorwould reduce the topography of the fitness landscape to a ‘golfcourse’, and render a computational study either infeasible orirrelevant to biology. Organisms certainly benefit from a lessthan perfect clock.

Evolutionary computation has to be generalized tooptimize an entire class of fitness functions. For adaptivecircuits we can solve this generalized problem and showthere exists a path in the parameter space of our networksthat continuously improves both components of the fitnessfunction.

1.2. Adaptation in biology

It is convenient to describe adaptation as a feature of genenetwork that converts an input, e.g., a ligand, into an output,e.g., a transcriptional response. Adaptation occurs when anetwork registers a transient response to a rapid jump of inputbetween two nearly constant states. Perfect adaptation ensueswhen the output returns to precisely the same value wheneverthe input is constant, no matter what its value.

A classic example of adaptation in sensory transductionis the Ca-mediated desensitization of the photoreceptors invertebrates [5, 6] and allows us to perceive gradients overa wide range of ambient illuminations. This is not perfectadaptation in that we still sense the average light level.The MAPK pathway when stimulated by a step in the EGFconcentration will generate a pulse in active ERK and thenreturn to near-background levels [7]. Both processes aredriven by negative feedback [8]. Bacterial chemotaxis exhibitsperfect adaptation, in that the tumble frequency is independentof the mean chemo-attractant level [9]. Adaptation is usefulto the organism in that it extends the range of the sensory

Figure 1. Illustration of input and output relationships for a generalnetwork. The (external) input of the network is displayed in dashedred while the output is in green.

system. A quantitative analysis of biologically plausibleadaptive circuits was presented in [10] along with a refinedcharacterization of their dynamics. An evolution simulationasks the converse question, what networks are preferred ifcertain criterion are imposed.

1.3. Mathematical definition of adaptation

We simulate networks of interacting genes and proteins as asystem of differential equations. To evolve an adaptive circuitwithin this system, we designate a prescribed time-dependent‘input’ concentration, and a second ‘output’ concentrationfrom which the fitness is derived. The first requirement forperfect adaptation is that for constant input I, the output shouldtend to a constant O(I); so any fitness function should penalizepersistent time dependence.

Next, fix the concentration of the input to I1, far in thepast so O assumes a steady state O(I1), and at t = 0 changeI to I2, and follow O till it reaches a new steady state O(I2).For adaptive networks, the deviation

!Oss = |O(I2) ! O(I1)| (1)

should be small and ideally zero.Although !Oss = 0 is necessary for perfect adaptation,

it is not sufficient, since the trivial network where O isindependent of the input satisfies the condition. So we alsohave to impose that O varies if the input changes. We quantifythe change in O after the input I switches from I1 to I2 at t = 0,in terms of

!Omax = maxt>0|O(t) ! O(I1)|. (2)

This quantity defines the response of the network to achange of the input. We do not care about the sign of theresponse or whether it is monotonic. Intuitively, a biggerresponse should correspond to a better fitness. More generally,when our model input consists of a series of plateau, (1) isthe difference in O between the first and last time on eachplateau, and the maximum in (2) is taken over one plateau ata time. Both functions should be averaged separately over allplateau.

To sum up, a network is perfectly adapting if and only if

• it responds to the input: !Omax > 0;• its steady states concentration does not depend on the

input, i.e. !Oss = 0.

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Thus, evolving adaptation requires the optimization ofthese two quantities. Since fitness is one function we have tocombine these two quantities, e.g.,

fa = !Oss !!Omax, (3a)

fb = !Oss + "/!Omax, (3b)

fc = (!Oss + ")/!Omax, (3c)

fd = (!Oss + "Oav)/!Omax, (3d)

where " is a fixed number, small in comparison with typicalvalues of O, and Oav is the average value of O over the time ofsimulation. Note by analogy with statistical physics we havedefined the fitness as a cost function to be minimized.

Any reasonable fitness function such as those in (3) favors!Oss small and !Omax large but assumes different tradeoffsbetween them. For example, an evolutionary path that ismonotone decreasing for one fitness function may not bemonotone for another, even if the optima are the same.

Still, some fitness functions are more meaningful thanothers. For instance, (3a) and (3b) are clearly dependenton the absolute value of the output. In the limit " " 0,fitness (3c) is dimensionless and therefore is less arbitrary. The" term has been added to prevent an ambiguous ratio of 0/0.Once !Oss = 0, the " term pushes !Omax to higher values.Fitness (3d) is also dimensionless, but tends to select for bigresponses relative to the average concentration of output Oav.For random networks relaxing exponentially toward a steadystate, !Omax # !Oss , so that a ratio fitness, such as (3c)and (3d), correctly assigns a (maximum) value 1 to all suchnetworks. Later in evolution, any increase of !Omax relativeto !Oss will be selected.

2. Results

Our evolutionary algorithm was described previously [2]. Itmodels transcription as a Hill function with three parameters:a maximum rate, a binding affinity between transcriptionfactor and promoter and an exponent defining the nonlinearity.The steps between transcription and the appearance of newprotein are modeled as a time delay [11, 12]. Proteinscan form complexes as defined by forward and backwardrates. We define a phosphorylation interaction betweenkinase and substrate by a Hill–Michaelis–Menten forwardrate (with an exponent to model cooperativity) and a constantdephosphorylation rate. Explicit formulae are given in theappendix.

Mutations can create or remove genes, change thetopology of the interaction network by adding or removinginteractions or change parameters of existing genes andinteractions. A ‘population’ consists of typically 50 differentnetworks. After one round of mutation, the most-fit half ofthe population is copied and replaces the remainder, and theprocess repeats.

The input as a function of time is a series of level plateau,whose values are log-normally distributed and whose averageduration imposes an upper bound on the timescale on thenetwork. The metrics !Oss and !Omax are averaged over

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Figure 2. Typical evolutionary trajectories for fitness functions(3a), (3b) and (3d) from top to bottom. The fitness contour lines aresmooth and equally spaced with red high and blue low as labeled.A typical evolutionary trajectory is shown in green evolving towardlow values and ending at the blue cross. Almost perfect adaptivenetworks are evolved in 400 generations with similar topologies forall the three functions.

all the plateau and used in equations (3). There is an overalltimescale in our simulations that will adjust to fit the responsewithin the duration of the steps in the input time course. Thisfeature of the environment is directly mapped into the evolvedparameters by any of our fitness metrics.

Similar results were obtained for all fitness functions in(3) as illustrated in figure 2. We next discuss the two classesof networks obtained and then show that they do not dependon the fitness function, mutation rates or the statistics of theinputs.

2.1. Type I: buffered output type

Example: figure 3

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Figure 3. Network topology of type I. Sketch of the topology of the network with the typical behavior of the output (green) under thecontrol of a random input (dashed red) as a function of time (parameters for this simulation are # = 1, " = 2, $O = 1, d = 0.5, $C = 0 andf (O) = O). Only the output undergoes constitutive degradation in the limit of perfect adaptation.

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Figure 4. Network topology of type II. Sketch of the topology of the network with the typical behavior of the output (green) under thecontrol of a random input (dashed red) as a function of time (parameters for this simulation are # = 1, " = 0.1, $O = 1, d = 0, $R = 0 andf (R) = R).

O = # ! "f (O)I + dC ! $OO, (4)

C = "f (O)I ! dC ! $CC, (5)

where f is an arbitrary function describing the coupling of theinput to the output (f (O) = O for protein–protein interactionsor f (O) = On

!"On

0 +On#

for catalytic interactions; both kindof networks were obtained) and " is our generic symbol forthe strength of interaction. Selection forces $C to zero.

2.2. Type II: ligand–receptor type

Example: figure 4

R = # ! "f (R)I + dO ! $RR, (6)

O = "f (R)I ! dO ! $OO. (7)

Computational evolution sets $R to zero.

3. Why are these topologies selected?

We need to show both that the final network is adaptiveand explain why the evolutionary algorithm is able to findit independently of the mutation parameters. Let us considera sufficiently general equation of the type I networks:

O = # ! "OI + dC ! $OO, (8)

C = "OI ! dC ! $CC. (9)

Now, in this problem, there is a clear correspondencebetween the physical parameters in the equation and the twometrics comprising the fitness functions:

• !Omax > 0 iff " > 0: there must be an interactionbetween O and I to make the output depend on the input;

• at steady state, Oss = #

$O + $C "I

d+$C

, so that !Oss = 0 for all I

iff $C = 0 or " = 0.

Any plausible fitness function for this problem shouldincrease !Omax: it is therefore obvious that evolution should

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Figure 5. Variation of !Omax and !Oss with parameters for the type I networks. The left panel shows the parameter sampling used in thesimulations with color indicating values of $C as seen in scale bar. Smaller sizes correspond to smaller values of $C . The right panel showstypical trajectories while varying only one parameter, $C , at fixed epsilon (dashed, " = 0.1, 0.3, 0.5, 1) and " at fixed $C (solid,$C = 0, 0.1, 0.3, 1), the other parameters being held constant. Arrows indicate the directions of increasing $C and ".

choose " > 0. In other words, this interaction and thus thenetwork topology will be selected as the first step of evolution.

It can also be seen that the fitness should decreasemonotonically as $C decreases to 0 from the expression forOss when expanded for small $C :

!Oss # #

$2O

$C

d"|!I |.

In the opposite limit of large $C, C will be close to 0, soa quasi-equilibrium value of C can be substituted into the Oequation:

O = # ! "OI ! $OO. (10)

It is clear from this equation that O simply relaxesexponentially toward its steady state, so we have

!Omax = !Oss # #"|!I |($O + "I )2

. (11)

Since by definition !Omax ! !Oss , the equality in (11)clearly corresponds to a (bad) limiting case; any move thatbreaks the equality of!Omax and!Oss should be evolutionaryfavorable. Thus, selection must decrease $C to decrease!Oss

(and increase the relative value of!Omax). This is also seen onthe left panel of figure 5 when all parameters are randomized,adaptive networks are obtained only for low values of $C .

We can also compute an estimate of the optimum!Omax.Let us suppose that $C = 0. The network will have a maximumresponse if O relaxes quickly ($O + " $ 1), while C relaxesvery slowly (d % 1) toward its steady state value. Shortlyafter the jump from I1 to I2, when O reaches its maximumOmax, C has not moved appreciably from its pre-jump value,C1. Using Oss = #/$O the stationary value of O, one has

0 = # ! "OmaxI2 + dC1 ! $OOmax

= $OOss ! "OmaxI2 + "OssI1 ! $OOmax,

and one gets

Omax = Oss

$O + "I1

$O + "I2, (12)

!Omax # !Oss

"

$O + "I2!I. (13)

Note that !Omax is an increasing function of " under ourassumptions. However, the typical timescale of response willbe controlled by d, which cannot be too slow if we want aquick response (i.e., within the duration of the plateau I2).The statistics of the plateau duration enter here since a larger!Omax may be realized by tolerating an incomplete responsefor very short plateau so that d can be smaller and increase theresponse from more typical plateau.

From the right panel of figure 5, we can see that for typicalparameters there is no tradeoff between increasing!Omax anddecreasing !Oss : increase of !Omax is not only relative to!Oss , but also absolute all other parameters being fixed, i.e.by decreasing $C , one moves toward the upper-left corner ofthe plane displayed in figure 5.1 This is the reason why anyfitness function tending to maximize!Omax while minimizing!Oss should be able to select for this network.

From all these considerations, one can predict that(figure 5)

• evolution will select the topology first since" > 0, !Omax > 0, and thus create the otherparameters on which selection can act,

• once the topology is selected, evolution should decrease$C , since !Oss = 0 iff $C = 0 and since decreasing $Cboth increases !Omax and decreases !Oss ,

• evolution should increase " and adjust the various ratesto maximize !Omax, in response to features of the inputsuch as its typical timescale.

Similar arguments can be given for the other networktopology (figure 4) that evolved for which a sufficiently generalsystem of equations is

R = # ! "RI + dO ! $RR, (14)

O = "RI ! dO ! $OO. (15)

Now, in a similar way

• O is defined as the reaction between R and I so trivially!Omax > 0 iff " > 0, and there will be a response of Oto I;

1 The increase in !Omax when $C decreases can made plausible by takingthe ratio between equations (11) and (13), to yield 1 + "I/$O which is alwaysbigger than 1.

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Figure 6. Variation of !Omax and !Oss with parameters for the type II networks. The left panel shows the parameter sampling used in thesimulations for different values of $R . The panels follow the same conventions as in figure 5.

• at steady state, Oss = #

$O+ $R(d+$O )

"I

, so that!Oss = 0 for all

I iff $R = 0.

Given the interaction, it is very easy to evolve parametersto get an adaptive network: as before, it is clear that !Oss isan increasing function of $R , so that one has to minimize $Rto minimize the fitness. When $R is large the R equation isat quasi-equilibrium, so R is a linear function of O and the Oequation describes exponential decay to its steady state. Soreducing $R also increase !Omax with respect to !Oss .

Now, if $R = 0, again one can estimate!Omax. Assumingthat O responds more rapidly than R, then R will still beapproximately R1 when O attains its maximum Omax andO = 0. So we can solve for Omax:

0 = "I2R1 ! dOmax ! $OOmax

= I2/I1(d + $O)Oss ! (d + $O)Omax,

i.e.

Omax = (I2/I1)Oss

or !Omax # Oss!I/I1.This represents a clear physical limit on the maximum

value O can reach for any set of parameters. Thus evolutionwill favor rapid O and slow R responses, which can be realizedby several different parameter combination. Again, there is nocompetition between minimizing!Oss and increasing!Omax

in figure 6.2

The shapes of the " and $R contour lines in figure 6 arepartially due to the breakdown in the quasi-static assumptionsmade above. For $R & 0, increasing " decreases the responsetime of R and thus allows !Oss more time to reach itsasymptotic value of 0. But a more rapid response of Rlimits the maximum excursion of O and thus !Omax. Forlarge $R , the curve parameterized by " is double valued over!Oss . The upper branch has the same rational as for small $R .On the lower branch for small ", the system behaves as if O isa function of I and thus !Omax and !Oss are proportional.

2 In the limit of big $R , it is clear from equations (14) and (15) that R, O andtherefore !Omax are zero so one has to decrease $R to have nonzero !Omax.

4. Networks with constraints on the types of inputs

The previous examples did not place any restrictions on thetypes of input, and the evolution selected an input that couldeither participate in a protein complex or act catalytically, theinput was never a transcription factor. What happens if weconstrain the input to act transcriptionally but let evolutiondefine all other parameters?

Again we found adaptation, with one predominanttopology shown in figure 7 along with the fitness as a functionof time.

The general equations of these network are

O = f (I)(t ! %O) ! "RO + dC ! $OO, (16)

R = g(I)(t ! %R) ! "RO + dC ! $RR, (17)

C = "RO ! dC ! $CC, (18)

where we make explicit the temporal lags %O, %R between theprotein concentration at the promoter and the appearance ofnew protein.

4.1. Why is this topology selected?

Since three interactions are needed to build this network versusonly one for the type I and II networks, it was necessary toconstrain the type of input to eliminate the latter two networks.It is less obvious why there is a downward path in fitnessfavoring it. As before, since evolution favors !Omax > 0,the first mutation will be for the regulation of I by O. Then aneutral mutation can create another protein R under the controlof I. We then have to show why a protein–protein interactionbetween O and R (i.e. " > 0) is favorable.

At steady state, one clearly has the followingrelationships:

0 = f (I) ! O

O + &g(I) ! $OO, (19)

with & = $R(d+$C)$C"

. For small " (large &), one therefore has atlowest order

Oss # f (I)

$O + g(I)/&# f (I)

$O

$1 ! g(I)

$O&

%,

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+

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ess

1200 1000 800 600 400 200 0

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Figure 7. Topology of evolved networks when the input is a transcription factor. Right panel: typical evolution of fitness (3d) for thissimulation. The first plateau occurs when activation of the output by the input is selected. When the protein–protein interaction appears, thefitness drops further and parameters are optimized to reach perfect adaptation.

so that

!Oss #&&&&

$f '(I )

$O! g(I)f '(I ) + f (I)g'(I )

$2O&

%!I

&&&& .

From this expression, one clearly sees that !Oss isreduced for " > 0 if f and g are both increasing ordecreasing functions of I. This is assumed by our algorithm(and accords with biological norms) since most transcriptionfactors (together with cofactors, and absent any covalentmodifications) act either as activators or repressors.

To evaluate Omax, we evaluate (16) and (17) in the quasi-static limit and assume for the lag times %O < %R , so that Oresponds before R does and returns O to the steady state (whenthe opposite inequality between the lag times applies, it canbe shown that !Omax = !Oss):

Omax = f (I2)

$O

$1 ! g(I1)

$O&

%,

(20)!Omax # f '(I )

$O

$1 ! g(I)

$O&

%!I

or

!Omax

!Oss

#1 ! g(I )

$O&

1 ! g(I )$O&

! g'(I )f (I )$O&f '(I )

. (21)

Because g'(I ) and f '(I ) have the same sign, !Omax >

!Oss . So this evolutionary move increases !Omax relative to!Oss . However, there is an implicit tradeoff in that !Omax

decreases from its value with " = 0. Both fitness functions(3c) and (3d) avoid this problem since they score just the ratioof !Omax and !Oss .

Once this topology was selected, we verified many timesnumerically that perfect adaptation follows on a decreasingfitness path (see figure 7 for a typical example). Againthis requires one species to be stable, i.e. $O = 0, and theparameters in g(I) and f (I) changed so that g(I) = µf (I).Then Oss = &/(µ ! 1). The response is much stronger ifthe time lag in the O equation %O = 0, while the lag in the

R equation is large. When combined with g(I) = µf (I) theinput to the O equation becomes a finite time difference (cfequation (19)). However, if the input does not vary over toowide a range, it is possible to have approximate adaptationif $O is nonzero for some specific values of f and g (seefigure 8 for examples of both situations). Plausibly a largerregion of parameter space is compatible with approximateadaptation than the exact variety.

5. Extensions to more general networks

Subnetworks with the topology of types I and II can evidentlybe found in more complex circuits, but there is rarely specificevidence that their parameters are in an adaptive regime.We have obtained elaborations of the basic types which canappreciably amplify the response. Figure 9 generalized thetype I network and obeys the equations

O = # ! f (O)I + dCC ! $OO, (22)

C = f (O)I ! dCC ! Ig(C) + dDD, (23)

D = Ig(C) ! dDD. (24)

A generalization of type II networks, figure 4, replaces theoutput by a transcription factor which drives the new outputspecies (figure 10). If the parameters in the transcriptionalinteraction are adjusted properly, the output can be made torespond to only upward jumps in the input.

5.1. General adaptive networks

Reference [10] classifies adaptive networks loosely as feedforward where one input activates two parallel pathwayswhich combine to generate an output, or feedback where theoutput negatively regulates itself. The feedback can eitherpromote the decay of the output or block its activation bythe input or both. Activation of a phosphatase is a common

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Figure 8. Behaviors of evolved networks from (16), (17) and (18). Left panel: dynamics for an example where non-linearities are differentin f and g. Parameters are f (I) = 0.6/(1 + I/0.54), g(I ) = 0.96/(1 + (I/0.7)1.4), $0 = 0.18, $R = 0.14, " = 0.9, d = 0.3, $C = 0.8,%O = 0.8 and %R = 16. Right panel: example where nonlinearities on f and g have evolved independently to be very similar. Parametersare f (I) = 0.4I 4.4/(0.674.4/ + I 4.4), g(I ) = 0.98I 4.2/(0.684.2/ + I 4.2), $0 = 0.01, $R = 0.09, " = 0.8, d = 0.4, $C = 0.8, %O = 0 and%R = 13.5.

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0.2

Figure 9. Amplification of the output from the type I network. Left panel: sketch of the network. Middle: dynamics for the networkwithout the interaction creating D. Right panel: dynamics of the complete network. (Input has been rescaled for display on both panels.)Parameters are # = 0.098, f (O) = 0.83O, dC = 0.85, $O = 0.44, g(C) = 0.76C1.75/(0.051.75 + C1.75) and dD = 0.75.

R

TF

Transcription

Input

Output

Out

put

TFTime

Con

cent

ratio

n

1400 0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0 200 400 600 800 1000 1200

Figure 10. Amplification of the output of a type II by an additional transcriptional step, TF. Left: topology used. Middle: typical behaviorof such a network (input in dashed red, TF in green, output in blue). Right: illustration of the mechanism of amplification. The steady stateconcentration of the output is displayed as a function of the TF level. The steady state of the system is low (blue cross), so that excursion ofthe TF into the nonlinearity (green arrow) triggers massive response of the output (blue arrow) relative to its steady state concentration.

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Phys. Biol. 5 (2008) 026009 P Francois and E D Siggia

way to deactivate pathways that signal by phosphorylationand receptor deactivation blocks further input after pathwayactivation.

Our network in figure 7 is loosely feed forward, the inputactivates the output and a second species, and the protein–protein interaction compares them. It actually implementsa finite time difference of the input, so constant inputsproduce no response. Most of the networks in [10] arebased on phosphorylation–dephosphorylation, so the totalconcentration of any protein is constrained which makes itmore difficult to achieve perfect adaptation.

Our type I and II networks are best understood as buffered,there is a continual production and decay of protein, pinningthe output to a fixed value irrespective of the input, providedit is constant in time. A change in input transiently affects theproduction or decay rates of the output and therefore elicitsa response. Output buffering is accomplished by the othervariable in the network which records the absolute level of theinput. In the type I networks C & I , while in type II R & 1/I .These networks parse the input into its derivative and meanamplitude.

A potentially biologically relevant distinction between thetype I and II networks is how they respond to large jumps ininput. The response of the former is &I2 ! I1 (in the limit oflarge $O), while the latter respond as &I2/I1, i.e. linear versuslogarithmic.

A generalization of networks of types I and II considersa set of species Oi that are inter-converted at rates governedby I, and which when summed satisfy

'Oi = # !

'$iOi. (25)

At steady state, we have($iOi = #, so this specific

weighted sum of Oi is adaptive. To capture this suminto a single variable would require implementing the decayprocesses by some common interaction (e.g. protein–proteincomplex formation). The concentration of the commonreactant would then itself adapt. Note that the concentrationof any Oi with a nonzero decay constant in equation (25) isuniformly bounded for all values of I. In the type I and IInetworks, the designated adaptive output was one of the Oi inthe sum, so the decay rate of the remaining variable had to bezero for adaptation. Thus nothing constrains the non-adaptivevariable to be uniformly bounded irrespective of input level,and as already noted C & I in type I, while R & 1/I intype II.

6. Discussion

There is a large literature on ‘digital life’, compact programsthat increase in copy number, mutate and complete forresources in a biologically inspired fashion [13, 14].Evolutionary simulations also purport to illuminate moregeneral issues of complexity, robustness and evolvability inbiology [15–17].

Our focus is much more on engineering, namely specificdifferential equation models that perform specific tasks. Forthe results to be biologically credible requires a demonstrationthat the fitness function is realistic and the mutation rate

parameters do not matter. The mathematical expression ofthe fitness function should be as general as possible and omitnumerical constants, other than those that define scales of timeand concentration.

To understand the space of plausible fitness functions, itis very helpful to parameterize it with multiple metrics. Forvision, both acuity and brightness play a role, but their relativeweight depends on the animals lifestyle [1]. For adaptationwe choose conditions (1) and (2). Additional less importantcriterion might include a rapid response, which we imposedimplicitly by randomizing the duration of the constant plateauin the input function, and limits on the total amount of proteinused in the network, which we ignored. We assume then thatfor a particular network, organism and environment the truefitness is some function of these metrics. Mutations thatimprove all metrics should clearly be accepted. Whentradeoffs are necessary, we need to pause and enumerate thelimiting networks for which no further improvements in allmetrics are possible. For adaptation this last step was notnecessary, both metrics could be satisfied simultaneously.

A single fitness function that may seem natural foradaptation just forces O to behave as I + cst , by means ofa dimensionless linear correlation

1 ! r(O, I ) = 1 !&&&&&

(OI ) ! (O)(I ))

((O2) ! (O)2)((I 2) ! (I )2)

&&&&& . (26)

Its deficiencies from our perspective are that it favorsan overly specific linear correlation between the instantaneousderivative of I with O at the same time. With a more continuousinput time course, we have seen that (26) gives rise to networksof types I and II, but the amplitude of the response was free,not optimized, and in many cases very small.

Once a fitness space is defined in terms of several metrics,we can argue for the irrelevance of mutation rates if there isa monotone decreasing path to the desired state. If there arebifurcations in the path, and the fitness improves along bothbranches, then both arms have to be followed, since in realitythe branch chosen will depend on the relative rates, which areunknown. On a serial path, the mutation rates only matterfor the overall time, which is of no interest. When positiveselection directs the parameter choice in a model, ‘parametertuning’ is no longer a pejorative, it is how nature works.

Incremental evolution and arguments about the efficacyof positive selection suppose small mutational changes in thegene network. To check our analytic arguments we haverun our simulations in this mode. But all adaptive networkswere also obtained by taking random parameters (within fixedbounds) for each mutation. We can then argue that adaptivenetworks occupy a nonzero volume of parameter space. Ofcourse a parameter cannot be set exactly to zero by randomsampling, but the network can get close enough to give agood fitness. Small genetic changes can have many variableeffects on parameters, e.g. there are dominant negative pointmutations.

We evolved small adaptive networks, e.g. (3), (4) thatcan be inserted into larger gene circuits in any context, butit is often not clear if they occur with parameters that renderthem adaptive. One case where rates are known are G-protein

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Phys. Biol. 5 (2008) 026009 P Francois and E D Siggia

receptor kinases [18, 19]. They are very specific for activatedreceptors, and lead to the binding of arrestin and subsequentendocytosis. Thus they fall into our type II class with $R = 0.If the cell receives stimuli with a repeat time much shorterthan the recycling time of the receptors, then there will be anaverage steady state and the recycling process would satisfyour assumption of constitutive production (e.g., the # term in(14)). However, one instance of G-protein signaling that hasbeen quantified does not exhibit adaptation [20].

Prior work on adaptive networks includes a large literatureon bacterial chemotaxis [21–25], which is much more complexthan anything we have evolved. Bacteria can chemotax overa huge, 105 range in ambient ligand levels and do so inpart by receptor clustering [26] which is not included in ouralgorithm (though we can evolve receptor dimerization prior toactivation). In addition, there is probably considerable signalamplification happening along the pathway from receptor tomotor, and molecular noise could well be an important factorin defining the network. We have only evolved deterministicequations here, but could easily generalize to stochastic ones.Other specific features of the Escherichia coli chemotacticnetwork, such as the four-receptor methylation states, are notnecessary for adaptation; two will suffice. It remains aninteresting question whether if we took the known networktopology and evolved just the parameters from a randomstarting point, adaptation would emerge. There are manyconstrained parameters in all existing models: for instancemethylation and demethylation rates are assumed independentof the methylation state, fully methylated receptor are alwaysactive, fully demethylated receptors are inactive, etc.

Soyer et al [4] evolved parameters with a fitness functioncorrelating the derivative of the input with the output (andtherefore similar in principle to (26)) for a phosphorylationnetwork with fixed topology. Iglesias and co-workers [27, 28]wrote down several quadratically nonlinear adaptive systemsin two variables that encompass our type I and II models incertain limits. They did not consider their evolution.

7. Outlook

More refined simulations of adaptive networks are possibleby imposing more criteria on the selection. One can modelstochasticity due to small numbers of molecules and thenimpose a cost based on the amount of protein used [29].In stochastic simulations, selection for amplification can becombined with adaptation, with obvious relevance to signaltransduction. One can also ask for adaption that minimizesthe rates of protein production and decay to which we did notassign a cost. The questions and approach remain the same:are there obligatory tradeoffs between the multiple criterion,and can an adequate solution to the problem be achieved bystepwise positive selection.

Evolution driven by selection is fast, as Darwinunderstood intuitively, and [1] made explicit for the eye.The alternative metaphor of multiple fitness peaks createdby epistasis makes transitions much more contingent onunknowable parameters implicit in moving through fitnessvalleys.

With the exception of [2], all efforts we are awareof to evolve simple genetic networks do not control forbiases implicit in the fitness function and the mutation rates.Adaptation has provided a clear illustration of how to treat afamily of fitness functions, and an example where mutationrates demonstrably do not matter for the final outcome.If pathways in developmental biology can be recovered ingeneral terms by our methods, it implies that evolution selectsfor structures that can be learned rapidly from the randomchanges supplied by mutation. Better structures that cannotbe reached incrementally are invisible. Evolution recast as asearch for networks that can be found by incremental selectionmay become a deductive theory.

Acknowledgments

This work was supported by an NSF grant DMR-0129848 toES, and Lavoisier Fellowship to PF. We thank J Skotheim forcomments on the manuscript, and John Guckenheimer andWilliam Bialek for discussions.

Appendix A. List of interactions with equations

The following table summarizes all possible interactions thatour algorithm can choose, with corresponding equations andranges of variation for parameters:

Authorizedrange of

Example of parametersinteraction Equations variation

Degradation ofprotein A

dAdt

= !$AA $A * [0, 1]

Unregulatedprotein Bproduction

dBdt

= #B #B * [0, 1]

Transcriptionalregulation ofprotein B byprotein A

dBdt

= #B1

1+(A(t!%B)/A0)nA0 * [0, 1]

n * [!5, 0] foractivationn * [0, 5] forrepression%B * [0, 20]#B * [0, 1]

Dimerization: Aand B form adimer C

dAdt

= !"AB + dC " * [0, 1], d *[0, 1]

dBdt

= !"AB + dCdCdt

= "AB ! dC

Phosphorylation:K phosphorylatesprotein A intoprotein A+

dAdt

= !"K (A/A0)n

1+(A/A0)n+ dA+ " * [0, 1]

dA+

dt= "K (A/A0)n

1+(A/A0)n! dA+ d * [0, 1]

A0 * [0, 1]n * [0, 5]

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Phys. Biol. 5 (2008) 026009 P Francois and E D Siggia

When multiple transcriptional activators are present, theircombined output is defined by the maximum of their activities.Repressors are combined multiplicatively. This is analogousto using an OR between activators and an AND betweenrepressors in discrete systems.

All these interactions are combined to create the finalequations. Suppose for instance that P is repressed by R,activated by A1 and A2 and phosphorylated by K, the P reactionrate would typically look like

dP

dt=

$max

*An1

1

An11 + (A1P )n1

,An2

2

An22 + (A2P )n2

+

, #P

1 + (R/RP )n3

%(t ! %P )

! "K(P/P0)

n4

1 + (P/P0)n4

+ dP + ! $P P . (A.1)

Appendix B. Algorithm overview

The strategy is to evolve genetic networks by repeated roundsof selection, growth and mutation, and is very similar to ourprevious works [2, 3].

Typically 50 different networks are followed in parallel.At each step of the algorithm, differential equationscorresponding to each network are integrated with randominputs as described in the main text. Then, the fitness functionis computed from the final state of each network.

After the fitness function is computed, the 25 bestnetworks are retained (selection), copied (growth) and mutated(mutation). Mutations are chosen randomly from among thefollowing types:

• creation or removal of an existing gene,• creation or removal of any regulatory linkages

(transcriptional regulations, protein–protein interactions,phosphorylations),

• change of any kinetic parameter,• change of the reporter proteins on which the fitness

function is computed (i.e. change of the output).

Kinetic parameters are changed by resampling uniformlyfrom a predefined range from 0 to a maximum of O(1) timesarbitrary constants defining the units of time and concentration.Mutations that change topology (add or remove a gene orinteraction) occur at one-tenth the rate of parameter changingmoves. As explained in the main text and described earlier[2], results are insensitive to mutation rates. The degradationrates in particular never converge to their upper limits.

After the mutation step, the entire process is iterated. A‘generation’ is one iteration of this selection/growth/mutationprocess. Since the creation of genes and interactions areseparate mutations, a gene may be created which has noeffect on the output and thus a neutral change. This genemay disappear in subsequent generations, or may be linked byan interaction to favorably affect the output. It will then befavored by selection and may ultimately fix in the population.

We used a simple Euler algorithm for time integration,to simply take into account transcriptional delays. The

integration time step was typically 10% of the minimumtimescale set by the kinetic parameters (i.e. for most casessince the range for degradation constants and most kineticparameters is [0, 1], the time step used is 0.1).

Specific tools have been developed in PYTHON to define,encode and modify genetic network structures and evolutionparameters. A PYTHON translator has been developed toautomatically write small C programs encoding the dynamicsof each network, compile and run it to integrate equations andcompute the fitness function. This computational strategy wasused to minimize the overall computational time for evolution.

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