Modelling evolutionary selective pressures

To simulate evolution in silico, I need to model selective pressures experienced by the evolving cellular networks and so link the contribution of a networks function to the overallfitness of the organism. Fitness is an abstract concept, representing the reproductive success of an organism and might be most tractable for microbes where it could be approximated by growth rate [178]. In BioJazz, the fitness of networks can be defined by the user, such that networks can be evolved under biologically motivated or artificial selective pressures.

The user-defined fitness function is used to evaluate the performance of a given network, encoded by a particular binary string, and to calculate a fitness score. In previous studies on the evolution of signalling and regulatory networks, the fitness function usually involved applying a stimulus to the network and evaluating its temporal or steady state response [36, 65, 123, 143, 179]. Different fitness functions relating to dynamical or structural features of the network can be easily constructed

as illustrated in the results section for ultrasensitive dynamics (additional sample files are included in the BioJazz web site) and adaptive dynamics (Figure 2.2).

Ultrasensitivity fitness function: Currently the fitness function used to score the ability of a given signalling network to generate an ad hoc switch- like function to mimic the ultrasensitive signal-response relationship evaluated the response to a three-step ramp-up and three-step ramp-down signal profile as shown in Figure 2.2A. For each ramp-up in the signal, the system is simulated to steady state before the next ramp is applied. The scoring function considered both the amplitude of the response to middle steps in ramp-ups and ramp-downs (amplitude scoreSamp) and the difference of the response amplitudes between the middle steps

and the other two steps (ultrasensitivity score Sult). If ymin and ymax are defined

as the minimum and maximum values of the response during the interval from a change in the signal to steady-state, then the response amplitude for each of the signal ramp-ups (indicated with a ’+’ sign) and ramp-downs (indicated with a ’−’ sign) is calculated as:

∆yi+=yi+max−yi+min (2.3)

∆yi−=yi−max−yi−min (2.4)

where the subscripts denote the corresponding ramps in the input signal. With these measurements, the amplitude score (Samp) is given as the normalized amplitude of

the change in response to the second ramp-up and ramp-down signals:

Samp =

(∆y2++∆y2−)/2

ytotal

(2.5)

withytotal being the maximum possible response (i.e. the concentration difference

between a fully active and fully inactive output protein), and acts as a normalisation factor ensuringSamp to be between 0 and 1. In order to quantify the ultrasensitivity

second ramp-up/ramp-down signals, and the first choice/third choice. I first define the difference in the response to the different ramp-up and ramp-down signals as

Su1 = (∆y2++∆y2−)/(∆y1++∆y1−) and Su3 = (∆y2++∆y2−)/(∆y3++∆y3−).

Then I can derive the ultrasensitivity score (Sult) as:

Sult= # ( Su1 ru+Su1 · ( Su3 ru+Su3 )) (2.6)

where, ru is a user-defined scaling parameter that ensures the two ratios Su1 and

Su3 (and thus the ultrasensitivity score) is between 0 and 1. Besides the amplitude

and ultrasensitivity scores, I also define a complexity score (Scom). It is plausible to

assume that networks are under selection to minimize their energetic burden to the cell, and this score allows us to capture network complexity. The complexity score is given by:

Scom=

rc

rc+C

(2.7) where C is the sum of the total number of rules, proteins, domains, and reactive sites in the ANC model and rc is a user-defined scaling parameter for scaling the

complexity score Scom between 0 and 1. Finally, the fitness function combines the

three scoring functions:

F = (Sωa

amp·Sultωu·Scomωc ) 1

ωa+ωu+ωc (2.8)

with theω_∗being user-defined parameters that control the weightings of the different scores.

Simple adaptation fitness function (Figure 2.2B): The key aspects of adaptive response dynamics are that the system shows an initial response to the input (∆O+max/−̸= 0) that the steady state value of the output returns to its pre-input

level, irrespective of the level of the input. In other words, after a sustained change in input (e.g. a step change), the output should initially respond but ultimately

Figure 2.2: Sample fitness functions for selection for networks with ultrasensitive or adaptive response dynamics. (A) The input signal (blue) used in the temporal simulations of the system for ultrasensitivity. Each ramp-up and ramp-down of the signal is introduced after the system reaches steady state. The corresponding system output over time is shown in green. The differences in steady state output between different signal levels, indicated as ∆y values on the plot, are used to calculate the amplitude and ultrasensitivity scores. (B) Illustration of the dynamics of input signal (blue) the output response (green) in simulations of the system for adaptive dynamics. The parameters in adaptive fitness function, ∆O+max/ and ∆O+ss/, are

labelled.

settle back to its original steady state: ∆Oss+/− ≈ 0. Therefore, the adaptation

fitnesswcan be configured as:

w= $ ∆O+max C · K K+∆Oss− · $ ∆O−max C · K K+∆Oss− (2.9)

whereCis a normalization factor to scale∆O+max/−and∆Oss+/−in to [0,1], andKis a

threshold parameter. By imposing such a selective pressure, it is possible to evolve an increased response sensitivity (∆Omax+/−) and reduced adaptive error (∆Oss+/−),

and so achieve networks with an adaptive response.

When the fitness function requires evaluation of the system dynamics, a temporal simulation of the network is executed by numerically integrating the set of ODEs arising from the interaction reactions in the network. To perform these simulations, BioJazz uses MATLAB⃝R with files automatically generated from ANCs

output via the Facile tool [180]. Stochastic simulation of the ANC model is also possible by customising the fitness scoring function.