Algorithm overview

φ-evo relies on an evolutionary algorithm acting on a population of networks, to explore the space of possible biochemical networks implementing a particular function of interest [14].

There are two levels in the algorithm: the individual level (white box on Fig 1C) where genotype, phenotype and evolutionary fitness of individuals are defined and computed, and the population level (green boxes on Fig 1C), where the evolutionary process is actually performed. All objects (biochemical interactions, networks, populations) are defined using customized Python classes, and explicit numerical calculations are compiled on the fly and performed in C for faster performance. A complete documentation for all Python classes enriched with examples is included and can be automatically (re)generated.

Individual level

Individual networks in a population are represented as Python objects called Mutable_Network. This class encodes the topology of the network considered, the different parameters corresponding to biochemical species and their interactions, and all the methods that will be used at the individual level, including fitness computation and mutations of the network.

Population level

We evolve a population of networks as defined above using a classical evolutionary algorithm. By default, rounds of selection, expansion and mutation are performed as depicted in Fig 1A.

At each generation, we compute the fitness of each network, and impose a strict elitist selection [19] where the worst half of the population is discarded, and the best half is selected and duplicated. Mutation operators act on every individual in the duplicated half population. Mutation rates are predefined by the user, and by default are rescaled at each generation to maintain on average a fixed number of mutations per network per generation (typically one or two). This prevents uncontrolled mutations due to combinatorial explosion, and fits the phenomenological Drake’s rule (a scaling relation of mutation rates with genome size) observed in real evolution [20].

We also include a Pareto version of our algorithm as proposed in [21]. In summary, Pareto evolution aims at optimizing simultaneously several constraints to circumvent the problem of building an ad-hoc fitness combining those. To rank the networks prior to selection, one sequentially computes so-called Pareto-ranks. The networks of rank 0 are those which are not dominated by any other network on all constraints. Rank 1 networks are only dominated by rank 0 networks, and so on. We then perform elitist selection based on ranks of networks. A fitness sharing option is also available to promote diversity in the population [21].

Since all individual networks can be integrated independently, φ-evo can be easily parallelized, and by default multi-threading is used. We routinely run φ-evo on modern multi-core machines or small clusters, with population of a few dozens individuals, so that each generation takes at most a few seconds of computation for examples presented here. Running time of the algorithm almost directly scales with the size of the problem studied, and in particular is proportional to the network size and to the number of cells simulated in an embryo. Small networks with few nodes such as the ones displayed on Fig 2 can evolve in less than a couple hundreds of generations and thus a successful run can be obtained in less than half an hour on a modern laptop (not all evolutionary runs are successful, but results presented here are “typical” in the sense that runs are more often succesful than not). More complex simulations have been done for embryonic patterning and typically require several thousands of generations to converge and give bigger networks (e.g. 10 to 15 node-networks were routinely obtained in [9]); this requires running times of several hours to a day.

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Fig 2. Visualization of the results of φ-evo.

(A,B,C) Application of φ-evo on biochemical adaptation, reproducing results of [22]. Panel (A) describes the fitness, similar to the one in [22]. For ideal adaptation, an output (yellow) should recover its steady state ΔO_ss → 0 after a change of the input (blue). To ensure that the input affects the output, one also wants to maximize ΔO_max. (B) presents the behaviour of a network evolved with φ-evo, where the output is subject to several changes of the input producing a perturbation of its concentration. The return to steady state is quick. Corresponding network topology is presented in panel (C) and is similar to solutions found in [22] (D,E,F) Application of φ-evo to segment formations, reproducing results of [23]. Panel (D) sketches mouse somitogenesis, coupled to embryonic elongation. We use input dynamics modeling elongation and a fitness function counting final number of segments similar to [23]. (E) shows the behaviour of an evolved network, where the system undergoes a bifurcation from an oscillatory to a bistable system when it exits the tail-bud (modelled as the region of high Input) (E-top). Depending on its state when leaving, the cell reaches one of two possible steady states. The constant rate of elongation then produces regularly spaced high concentration and low concentration cells (E-bottom). See [23] for more details. Corresponding network topology is presented in panel (F).

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Running

Evolutionary simulations in φ-evo are run via command line. C files corresponding to a specific evolutionary simulation (such as fitness, initial conditions, etc…) should be provided by the user. Evolutionary parameters (such as mutation rates, initial network) are encoded in a file called initialization.py. Those files should be put together in a common directory (called in the following Simulation) and an evolutionary simulation is started with

A known difficulty in evolutionary simulations is “code-bloat”, where combinatorial explosion and genetic drift can hinder the core structure of a working network. For this reason, we can also run new evolutionary simulations initialized with an evolved network of interest, while forbidding the addition of new nodes. This isolates the “core” working network, and allows for a better understanding of network dynamics.

Visualization

We have developed two Jupyter -notebooks to make φ-evo more user-friendly. The ProjectCreator notebook allows to define all simulation parameters (e.g. mutation rates) in specific widgets, as well as to define initial topologies for evolution if needed (e.g. to bias evolution with a specific network architecture). The AnalyzeRun Notebook allows to visualize results of simulations in a web browser. This is illustrated in Fig 2 with two simulations reproducing results from [22] and [23]. Users can identify interesting evolved networks, check their topology, simulate them on the fly, and do simple changes such as node removals.

lac operon

We first use φ-evo to recover the regulatory logic of the classical lac operon, discovered by Jacob and Monod [24]. The lac operon allows for regulation of the enzymes digesting lactose in E. Coli. It is activated when two conditions are realized: presence of lactose and absence of glucose. The lac Repressor (lacR) represses transcription of the operon while the CAP protein is responsible for its activation. When present, lactose is translocated by permease inside the cell and binds to lacR, which relieves the repression. In the absence of glucose, cAMP is produced in burst, then binds to CAP turning it into an activator. This regulatory mechanism is presented in Fig 3A.

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Fig 3.

A Scheme of the known regulatory network for the lac operon. The lac Repressor is inactivated by the presence of lactose which derepress the operon. The cAMP metabolite binds to the CAP protein to activate the operon. B Final network found by φ-evo after pruning of unnecessary reactions. The similarities with the network A are evident. The species with a bold contour are imposed by the definition of our initial network, all other nodes have evolved. C Time course of the network presented in B, the presence of both Inputs is indicated with the colored bars in the bottom. On the main figure, the output concentration is plotted against time while the desired output is indicated with a black line.

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The lac operon can be envisioned as a logical binary gate performing the (symmetric) boolean operation “cAMP AND lactose” between those two metabolites. To evolve this function, we initialize a network with an input corresponding to the lactose outside of the cell (denoted I₀), a permease-like enzyme translocating it into the cell, the cAMP species (I₁) and one output coarse-graining all transcriptional targets (O). (Bold contours for nodes in Fig 3B). Both inputs are boolean variables that can be be present (I_j = 1 in rescaled units) or absent (I_j = 0), and follow a random dynamics. Importantly we fix the function of the Inputs to be metabolites that cannot activate of repress transcription by themselves, and thus need to act through allosteric binding with other species.

To define the fitness function, we call T_on the total time when the output should be ‘on’ (i.e. when both inputs are present) and T_off the remainder where it should be off (i.e. at most one input is present). If we suppose that the output gives a benefit β during T_on and a cost γ otherwise, we compute the fitness: (1) and assume this function should be minimized. The min operator ensures that the output simply has to be higher than an arbitrary concentration rescaled to 1 to perform the computation. We also rescale φ by setting β = 1.

For convergence of the evolutionary process γ can not be arbitrary (see details in Supplement). Intuitively, if γ is big, it is very costly to get an intermediate step where one input alone controls the Output, which would impinge evolution. Conversely, if γ is small, there is no strong incentive to evolve a correct operon because turning the Output on all the time is not costly. We chose , and checked that the precise value does not change significantly the networks obtained when we vary it.

A representative result of a network found by φ-evo is shown in Fig 3B and the behaviour of the network is shown in panel C. We display the core network obtained after pruning unnecessary interactions as described in the Implementation section (See Supplements for more details). We can see by direct comparison that the final network of the algorithm appears very close to the structure of the real lac operon displayed in Fig 3A.

While this example might appear simplistic, it shows how even simple simulations can provide insights into the biochemical constraints on evolution. While the objective function puts the two Inputs on equal foot, the evolved network is not symmetrical, presenting regulations via one activation and one repression. This asymmetry actually originates from the default transcriptional grammar implemented: since multiple activators act via an effective “OR” gate, our algorithm can not directly build a transcriptional“AND” gate. The simplest solution is to have activation on one side and titration of the repressor on the other one, which is found by φ-evo.

Emergence of fitness in Pareto optimization

As a second example, we use φ-evo for Pareto evolution of embryonic patterning (Fig 4A). In [9], the computational evolution of hox-like patterning was addressed. A single combined fitness aimed at optimizing two terms: a global entropy term, maximizing the number of genes expressed within the embryo (F1) and a local entropy term, minimizing the number of genes expressed in a given cell to define local identity (F2), 4 B. The fitness F2 − F1, corresponding to mutual information between cell position and gene expression, was initially minimized, leading to the evolution of networks presenting several characteristic of hox-like patterning, including posterior dominance [9].

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Fig 4.

Pareto evolution of embryonic patterning (A) Schematic representation of the problem: we model embryonic patterning as an antero posterior sequence of genes at different positions (B) Fitness definition, where p(i; x) represents the relative fraction of gene i expressed at position x. F₁ increases with the number of genes expressed within the embryo, while F₂ increases with the number of genes expressed locally at x. See [9] for more details on the two fitnesses (C) Pareto fronts for a single evolutionary simulation at different generations (warmer colours correspond to later generations) aiming at maximizing F₁ and minimizing F₂ simultaneously. Inset corresponds to the evolutionary trajectory studied in panel E (D) Changes of functions F₁ and F₂ for random mutations within a network population (E) An evolutionary trajectory leading to the definition of a new Pareto front, from 3 to 4 domains. Only the local probability profiles for the steady states of the networks are displayed.

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Here we revisit this problem using multi-objective/Pareto evolution to minimize at the same time −F1 and F₂. Typical results of these simulations are presented in Fig 4C, where Pareto fronts for different generations are plotted.

The networks we obtain are qualitatively similar to the ones first evolved in [9] (see Supplement). Remarkably, it appears that the Pareto fronts observed during the course of evolution are slopes of constant F2 − F1, corresponding to the exact mutual information fitness used in [9]. During evolution the Pareto front clearly evolves towards increasing values of mutual information (corresponding to an increase of the number hox-like domains in an embryo [9]).

The choice of fitness function in [9] was in part motivated by the fact that gene duplications, presumably an important driver for evolution, were neutral moves for the chosen mutual information fitness (see Supplement of [9]). Here the logic appears inverted: during evolution, most mutations are (at best) expected to move networks on the Pareto front. This is illustrated in Fig 4D where the relative changes of fitness are displayed for a population of networks following random mutations. We clearly see that many of the mutated networks localize on the slope ΔF₁−ΔF₂ = 0 (in Supplement we generalize a calculation from [9] to explain this effect). This defines the equivalent of a neutral space, but for Pareto evolution: networks moving on the front are not Pareto-dominated by any other network by definition.

If most mutations are on the Pareto front, this raises the question of how a new Pareto front is reached during evolution. Such a (rare) event is presented in Fig 4E, where we have tracked back the ancestors of a network to build its evolutionary trajectory between two Pareto fronts (inset of Fig 4C). One clearly sees how this network incrementally evolves from 3 to 4 hox-line domains, by addition of an internal domain in the right most one, and refinement/shrinkage of all domains. Interestingly this evolution does not occur in a direction orthogonal to the Pareto front, rather the trajectory zigzags with an overall direction corresponding to a small variation in −F₁ (i.e keeping the global entropy constant) while significantly improving F₂ (the local entropy). This fits the idea that evolution here is driven by similar genes expressed in the same domain, then later specializing to define new domains. However, it is also clear that on the short evolutionary time-scale, evolution is not a straight line in a well-defined direction. We see rather complex stochastic dynamics with the interplay of the two constraints, as can be also gathered from the intermingled structure of Pareto fronts at different generations in Fig 4C.

This example thus illustrates how φ-evo can be used to study the dynamics of evolution of complex features, and how an effective fitness function can emerge from the simultaneous optimization of several constraints (the latter presumably being more generic in biology than the former [25]). Indeed the fitness rationalized a priori in [9] is discovered here a posteriori by φ-evo. We could imagine situations where the geometry of Pareto fronts varies during generations, so that the shape of the Pareto fronts and the corresponding effective fitnesses would vary as well during evolution. If the geometry of Pareto fronts is very different in different regions in terms of constraints optimized, we could even imagine local populations evolving in very different ways, potentially leading to numerical speciations.