Genetic Networks Described in Stochastic Pi Machine (SPiM) Programming Language: Compositional Design

If biological objects are created by natural selection, why are they composed of discrete modules? What has been the nature of mutations since the Darwinian epoch? This paper presents examples of genetic circuits in terms of stochastic π -calculus; a new mathematical language for nanosystems. The author used a constructor of five elements such as decay, null gate, gene product, and negative and positive gates. These primitives were applied to design genetic switches, oscillators, feedforward and feedback loops, pulse generators, memory elements


Introduction
Biological entities are different from artificial devices because they are developed by an algorithm that Charles Darwin called natural selection. Evolutionary simulations demonstrated that modular structures are rare and less optimal than fully wired counterparts (Poli et al., 2008;Thompson, 1998). That is why the masterpiece of biological networks is an entanglement. However, there are many examples of modular design in evolution; body plans of invertebrates and vertebrates, and modular metabolic networks within bacteria (Kreimer et al., 2008;Parter et al., 2007). Modular structures can spontaneously emerge if the environment changes over time (Kashtan and Alon, 2005). Modularity can also dramatically speed up evolution . On the other hand, modularity is a basis of our ability to divide a problem into parts and to scale up to large systems by a composition of elements.
The motivation of this experiment is a biologically inspired bottom-up design from basic primitives or modules which resemble the natural evolution process of a composition of gene clusters and protein domains (Kuznetsov, 2009a). Artificial genetic networks of increasing complexity were constructed from their components. An attempt to study in a formal way the relation between genotype and phenotype was done and the behavior of genetic circuits of different topologies was investigated. Special attention was given to using an adequate language to describe a large system with interacting components. In general, π-calculus is such a language. Modules were described in the π-calculus formalism because it is subsequently expected to explore a larger system of the modules, like genetic networks. The main point here is a modular nature of π-calculus that opens the door to the combinatorial design on the basis of Stochastic Pi Machine (SPiM) programming language.
Basically, π-calculus is a model of computation for concurrent systems whose configuration may change during the computation in which "everything is a process" and all computation proceeds by communication on channels. The syntax of π-calculus lets us represent parallel processes, synchronous communication between processes through channels, and nondeterminism. A process is an abstraction of independent threads. A channel is an abstraction of the communication between two processes. Processes interact with each other by sending and receiving messages over channels. The content of messages is also channels. As a result of such a communication event, the recipient process may now use the received channel for further communication. This feature, called mobility, allows the network "wiring" to change with interaction. Stochastic π-calculus is an elegant approach to simulating the populations of biological molecules. A system is defined by a flow chart that gives the rates of transitions between states, for example the rate of protein production. The simulator chooses among the possibilities at random simulation. The SPiM is a programming language for designing and simulating computer models of biological processes. The language is based on π-calculus formalism and the simulation algorithm is based on chemical kinetics theory. The language can be used to model large systems incrementally by directly composing simpler models of subsystems. Some historical facts in the development of π-calculus should be mentioned: the π-calculus was invented in 1992 by Robin Milner with the aim of describing interactive systems. He showed that λ-calculus can be expressed in the π-calculus formalism that demonstrates a computation equivalence of πcalculus and a Turing machine (Milner, 1992;Milner, 1999). The π-calculus has been used to describe many different kinds of concurrent systems from mobile nets to business applications. Aviv Regev and Ehud Shapiro used stochastic πcalculus for a representation and simulation of molecular  (Priami et al., 2001). Various π-calculus languages with particular formal syntaxes and semantics, as well as simulators for parallel processes were developed; the stochastic π-calculus was originally proposed by Corrado Priami in 1995, while Andrew Phillips and Luca Cardelli developed SPiM language and designed a stochastic simulator (Stochastic Pi-Machine, SPiM) that performed a Gillespie algorithm (Gillespie, 1977). The machine has been formally specified and approved for the stochastic π-calculus (Phillips, 2009b). The SPiM simulator is available for free distribution on different platforms (Linux, MacOS X, Windows). SPiM Player, a graphical user interface to SPiM was written by James Margetson and co-workers in F# language. Molecular systems were described via these tools as collectives of interacting automata in artificial chemistry (Cardelli, 2009). Recently, the SPiM calculus was promoted as a formal visual programming language for biology (Phillips, 2009a).
With this background, the aim of this paper was to investigate abstract gene-regulated networks in terms of basic primitives and their interactions. Following (Blossey et al., 2006), the SpiM simulator based on stochastic π-calculus was used for the modeling of concurrent biochemical processes during the bottom-up design of genetic networks.
The paper is structured as follows. The basic circuit primitives in SPiM calculus are defined in the "method". The section "results" describes the genetic circuits of increasing complexity such as neg and pos gates, circuit connections, switches, oscillator motif and repressilator, feedforward and feedback loops, memory elements and bifans. Genetic gates are defined in the SPiM programming language and showed in bold in the text and figures. In addition, the behaviors of gates are described through programming code in the "appendix" and the "supplement". The main contribution of the paper -valid modular design of genetic networks in SPiM language, as well as novel aspects of design are highlighted in the "discussion". For instance, the design job in SPiM programming language looks like an evolution process. Genetic gates with multiple inputs allowed us to develop genetic networks with a high connectivity. A robust memory element consisting of four neg genetic gates was successfully created. The stochastic noise in SPiM calculus demonstrated a novel phenomena. Some of the data is included in the "supplement".

Method
The Stochastic Pi-Machine Version 0.05 (Phillips, 2007)  A gap between the abstract mathematical model and the programming code, i.e. between the specification and the program that implements it was eliminated by compression as in earlier studies (Blossey et al., 2006;Blossey et al., 2008). Interactions between biological molecules such as DNA and protein transcription factors were considered as channels. Each element of the network is a genetic gate and defines an input/ output relationship corresponding to the binding and synthesis of transcription factors (Blossey et al., 2008). Network elements become autonomous and only the input/output relations determine their wiring. Unary reactions, i.e. protein degradation are followed by a stochastic delay. Binary reactions are represented as a channel with an input (?) and output (!). A protein reactant is defined as an input or output on a given channel. Extra notations are used in the paper such as the constitutive expression rate ε, protein degradation δ, gene unblocking η, and for the binding of the transcription factor. For instance, a transcription factor can decay in the reaction δ, a blocked gene can be unblocked by the reaction η. A gene can also produce a new protein independently in the reaction ε, where a vertical bar (|) represents parallel execution (see definitions below). An initial population of proteins is empty; they are expressed constitutively and stochastically by genetic gates.
A stochastic nature of molecular interactions on a nanoscale was described according to (Blossey et al., 2006). For example, τ ε is a stochastic delay where τ is a symbol indicating event delay, and ε is the stochastic reaction constant which gives the probability per unit time that the delay action will occur (Gillespie, 1977). When an action with rate ε is enabled, the probability that it will happen in the period of time t is P(t) = 1-e -εt . This distribution satisfies Markov's property of stochastic dynamics.
The following primitives in SPiM calculus were used to build networks with different topologies: 1) decay or degradation of a transcription factor tr(b) ≡ ≡ ≡ ≡ τ δ , 2) null gate or constitutive transcription null(b) ≡ ≡ ≡ ≡ τ ε . (tr(b) | null(b)), 3 Genetic networks with an increasing number of nodes and connectivity were investigated with a nominal set of parameters: the protein binding r = 1.0 and protein decay δ = 0.001 for any gate, the constitutive transcription ε n = 0.1 and repression η n = 0.01 for a negative gate, and the constitutive transcription ε p = 0.01 and activation η p = 0.1 for a positive gate. To exploit a parameter space, I gradually varied the rates r, δ, ε and η in the experiments. Algorithms were mapped to executable program codes for SPiM (see Appendix and Supplement 4).

Behavior of basic gates
The basic neg and pos gates were represented previously by Blossey and co-workers (Blossey et al., 2006). In the case of a negative gate (4) without input shown in Figure 1a, the gene can express a protein by performing first a stochastic delay at the rate ε n = 0.1 and then executing a new protein in parallel to the gene itself. Alternatively, in the presence of input ( Figure  1c), it can be blocked by an input on its promoter region a and then be unblocked during a stochastic delay at the rate η n = 0.01 to activate the gene. In contrast, the modified positive  genetic gate (5) depicted in Figure 1b, can follow in stochastic delay at the low rate ε p = 0.01 to constitutively express a new protein in parallel to the gene or it can carry out an input on its promoter region a and then activate transcription at the high rate η p = 0.1 ( Figure 1d).
Simulations were started in the absence of proteins by doing a constitutive transcription. The number of protein molecules initially increased and finally leveled off at equilibrium between the production and degradation depending on the parameters ε and δ. The constitutive expression and the output were higher for the negative regulator than for the positive one (Figure 1a, b).
To observe a response of genetic elements, an input allowed linear increases from 0 to 100 individual molecules, then decreased linearly to 0. The input a-molecules were injected into the system at a certain time to shape the curve b. As a result of the reaction, the negative gate behaved like an inverter (Figure 1c) whereas the positive gate increased the output signal about almost 10 times (Figure 1d). For each plot, the abscissa indicates the time of simulation and the ordinate is the number of molecules.
Apparently, the simplest circuit is the single gate interacting with itself in a feedback loop. In this case, a promoter region a of the gene is parameterized together with the transcribed protein tr(a). The transcription factor can repeatedly make an output acting on the promoter region a, or it can decay at the rate δ according to expression (3). If output was wired to input, then the negative and positive gates showed an autorepression and an amplification respectively (Figure 1e, f). The neg(a,a) gate stabilized at an appropriative low level of tr(a). In contrast, a magnitude of tr(a) for the pos(a,a) gate was about 100 times higher in my experiments.

Connection of components in networks
Once basic elements are defined, genetics circuits can be assembled by providing interaction channels connecting various gates.   ments that followed increased very rapidly. In contrast, a sequence of neg gaits demonstrated an interchange of outputs. In the absence of an initial input a, the odd gates showed high output values b, d, f. At the same time, outputs of the even gates c, e stayed at a low level (Figure 2b). A signal moved through the chain of neg gates with a delay depending on the parameter η n . The smaller the value of η n , the longer the time for gene activation and the slower the propagation of the activation wave (data not shown).
The same circuit including a negative feedback on the head gate showed a rather altered behavior. The performance was chaotic at parameter η n = 0.01. Gene activities fluctuated significantly by time and amplitude (Figure 2c). When the repression delay was increased to η n = 10.0, the system stabilized itself at a fix-point (Figure 2d). These results comply with Blossey and co-authors (Blossey et al., 2006).

Switches
The combination neg(a,b) | pos(b,a) to neg(a,a), it has two outputs a and b. In the absence of stimuli, the system can stochastically start with an occurrence of protein a before stabilization ( Figure  3a). A long-time simulation revealed spontaneous fluctuations in the expression of this protein, up to 126 molecules in a spike (Figure 3a, insertion). neg(a,b) | neg(b,a) is a bi-stable toggle switch, which can start up in one state or another and as a rule stay there (Figure 3b, c).

The composition
More interesting is a tri-stable toggle switch neg(a,b,c) |  neg(b,c,a) | neg(c,a,b) that can produce three alternative combinations of proteins, such as (a,b), (b,c) and (c,d). Each gate has two promoter regions and one output. They are wired to each other as shown in Figure 3d. Occasionally, the system did flip-flops from one state to the other at the conventional set of parameters (Figure 3f, insertion). When a repression by protein binding r was increased to the value 10.0 instead of 1.0, the toggle switch showed stable behavior for a long time. Thus, the tri-stable toggle switch demonstrates a possibility to build nontrivial genetic networks with high connectivity.

Oscillator motif and repressilator
An oscillator motif includes a negative feedback loop and a positive self-activation loop. It can be written in SPiM language such as pos(a,a) | pos(a,b) | neg(b,a). Preliminary experiments with standard values of parameters did not reach oscillations. Proteins fluctuated due to inherent stochastic noise in the system (Figure 4a). When the rate of positive gene unblocking was decreased to η p = 0.001, irregular oscillations of protein a were induced (Figure 4c).
A repressilator consists of three neg gates that mutually repress each other: neg(a,b) | neg(b,c) | neg(c,a). Simulation of the repressilator at nominal parameters resulted in an irregular duration of protein cycles (Figure 4b). Since dynamics of the network depends on relative rates of the parameters r, δ, ε and η (Blossey et al., 2008), it was possible to fix the value of one parameter (ε n = 0.1) in order to study the effects of the others. The decreased rate of gene unblocking η p = 0.001 and the increased protein binding r = 10.0 allowed an improvement of the regularity of oscillations. Populations of proteins stabilized near 100 molecules in each cycle with the duration of impulses being about 625 units (Figure 4d).

Feedforward loops
A feedforward loop (FFL) is a circuit from three genes that express two transcription factors one of which regulates the other and together they regulate a target gene. The FFL has eight possible subgraphs, however, only two of them, namely the C1 coherent feedforward loop (C1-FFL) and the I1 incoherent feedforward loop (I1-FFL) are highly abundant in transcription networks of E.coli and yeast (Alon, 2007). These motifs were denoted as follows: C1- FFL -pos(a,b) | pos(a,c)  | pos(b,c) and I1-FFL -pos(a,b) | pos(a,c) | neg(b,c). Variations in a constitutive expression of pos gates (ε p ) were critical for both feedforward loops. C1-FFL output at nominal parameters and an empty start condition is depicted in Figure 5a. Curves demonstrate a correlation between amounts of b and c proteins, depending on  pos(b,c) gate activity. The system stabilized at low records for protein b and a high quantity for protein c in agreement with the unitary regulation for b and double positive regulation for c proteins. When the value of ε p decreased, the amounts of both proteins dropped considerably. Spontaneous impulses of c protein expression were observed at the very small ε p = 0.0001, indicating a fast front and an exponential backside and amplitude up to 100 molecules (Figure 5b, c). When ε p went down to 0.00001, the time between impulses increased and they became far less frequent, and the system mostly slept (Figure 5d). This behavior is determined by very rare constitutive transcription events at pos(a,c) and pos(b,c) genes because of the enormous delay ε p . It may be that only a successful combination of activity for pos(a,c) and pos(b,c) gates with dependency on pos(a,b) gate is able to activate c output under the conventional conditions of relatively hard protein degradation δ = 0.001.
As usual, the standard set of parameters was used in experiments with input. Proteins a and b were injected at 500 and at 1000 time intervals of simulation according to a triangle-shape function. In the case of single input a, Figure 5e shows a sharp response of b and c proteins with saturation at 100 and 200 protein molecules respectively. A double injection did not change the shape of c output even during the simultaneous presence of both proteins a and b (Figure 5f). pos(a,c) and neg(b,c) gates (Figure 6b). When the value ε p increased, it meant an augmentation of constitutive transcription for the pos gates and production of both proteins b and c improved significantly (Figure 6a). In this situation, the role of the neg(b,c) gate was insufficient. On the other hand, when the constitutive transcription for pos gates decreased to ε p = 0.001, then the situation changed and a fine balance between pos(a,c) and neg(b,c) gates played a major role. As a rule, the productivity of neg(b,c) gate was enough to suppress c protein for a time. Warped flops in c output were observed as a result of chaotic neg(b,c) regulation activity (Figure 6c). When a value of transcription for pos gates decreased to ε p = 0.00001, the activation of neg(b,c) gate by pos(a,c) was very low and the system demonstrated a stable production of c protein once again (Figure 6d). Under these conditions, the neg(b,c) gene was able to inhibit c output only sporadically (Figure 6d, insertion). Subsequent simulations with inputs were performed at nominal parameters as before. An injection of protein a in the system provoked a fast coherent response for both b and c proteins up to 100 molecules (Figure 6e). Additional injection of protein b did not affect c output (Figure 6f). When injections were finished, the system retuned into an initial state.

Composition of feedforward and feedback loops (FFBL)
Coherent and incoherent FFBL were denoted as the pos(a,b)  | neg(b,a) | pos(a,c) | neg(b,c) and the pos(a,b) | neg(b,a) |  neg(a,c) | pos(b,c) accordingly. They operated at a nominal regime like coherent pulse generator and decoherent pulse generator ( Figure 7).   to extremely diverse behaviors in experiments. The rates of parameters ε n , η n , ε p and η p were changed gradually from 0.1 to 0.0001. Both FFBLs were investigated in detail during short and long simulations (up to 5 . 10 4 and 10 6 time intervals). Results of the experiments are in Supplements 1 and 2.
I discovered that the coherent FFBL circuit can be in four different states corresponding to particular patterns of protein expression: I -low basal production of b and c proteins, IIintensive stable production of b and c, III -spontaneous synchronous outputs of a and c, and IV -exhaustive expression of a and c proteins with gaps. These essential states were sig-nificantly influenced by stochastic fluctuations. Nevertheless, I did not observe any transition from one state to another when the parameters were fixed. A system with a standard set of parameters was typically in state I which demonstrated an equilibrium level of b and c proteins near 10 molecules and synchronous pulses of expression up to 100 molecules (S1: pictures 1, 6, 10, 13). When the value of parameter ε n was progressively decreased from 0.1 to 0.0001, then the frequency and amplitude of these pulses fell; the system reached a still state with low expression (S1: 1-4). Hence, a silencing of the constitutive expression for negative regulators reduced the When the value of η n was altered from 0.01 to 0.1, then the system generated more pulses with the amplitude up to 100 molecules (S1: 5). A shift of η n from 0.01 to 0.0001 led to a low protein expression of about 10 molecules (S1: 8) as in the previous still state. Small values of parameters ε n and η n determined the same output effects mediated by neg gates (S1: 4 and 8). Obviously, a rare unblocking neg regulation has a similar effect as the low constitutive expression.
When I investigated the properties of pos gates, the value of ε p increased to 0.1 and an expression of b and c proteins increased to 100 molecules; it is just a state defined as II (S1: 9). A production of those proteins significantly improved as a result of a more effective pos(a,b) and pos(a,c) constitutive expression. In contrast, when the ε p decreased, the frequency of synchronous positive pulses increased (S1: 11,12).
Activities of the coherent FFBL were more surprising when the parameter η p changed. The system occurred in an unruffled state I at value η p = 0.01 with a low level of expression for b and c proteins (S1: 14) exactly like the small rates of ε n and η n = 0.0001 (S1: 4,8). However, the system stayed in a new state III at η p = 0.001, producing proteins a and c synchronously instead of b and c. From time to time, the circuit generated complex pulse trains of a and c proteins with an amplitude of up to 100 molecules (S1: 15). A poor basal production of protein b at an average of 10 molecules interchanged with these islands for an intensive expression of a and c proteins. When the η p had a rate 0.0001, then the system occupied the next state IV with a rich synchronous output of a and c proteins and negative pulses to a low content (S1: 16). Thus, the progressive decreasing of probability of activation for positive   gates pos(a,b) and pos(a,c) led firstly to the stabilization of b and c expression at a low level (S1: 14), but later provoked an intensive production of protein a instead of b through a feedback loop. In the case of infrequent activations of the pos gates, the system rearranged itself from state I to states III and IV (S1: 15,16). This system generated synchronous pulses with positive and negative amplitudes for b and c proteins up and down to 100 molecules accordingly (S2: 1). When the rate of ε n was shifted from 0.1 to 0.01-0.0001, the system followed in state II. In this case, the constitutive expression of neg(b,a) and neg(a,c) genes decreased. A negative control of protein a dropped, hence protein a suppressed c. The output of c decreased from 200 to 100 molecules. In addition, the frequency and amplitude of b pulses fell (S2: 2-4).
When I increased the value of η n up to 0.1, the frequency of b and c pulses rose as a result of frequent unblocking actions for negative gates (S2: 5). When the value of η n decreased to  pos(a,b) and pos(b,c) genes leading to a superproduction of b and c proteins. I identified an incoherent FFBL circuit in the basic state I at ε p = 0.01, 0.001 and 0.0001 even though the frequency of pulses progressively increased (S2: 10-12).
Finally, when the value of parameter η p decreased to 0.01, the system woke up and changed from state I to state II (S2: 13,14). When η p decreased, the behavior transformation was more dramatic. The system turned on to state IV at η p = 0.0001. An expression of protein a appeared at an intermediate level up to 100 molecules and the expression of protein c decreased drastically to about 10 molecules (S2: 16) which I had not observed before. In addition, I discovered very interesting reactions at η p = 0.001: the system switched from time to time from state II to state IV producing intricate pulse trains of the protein a or c (S2: 15). I defined this state as III based on the instability and sufficient role of specific η p activation within the feedback loop pos(a,b), neg(b,a).
In general, both coherent and incoherent FFBL circuits produce coherent and decoherent pulses, being in state I at a nominal set of parameters. In addition, they behave as stochastic pulse generators in the instable state III at the particular low rate η p = 0.001.

Bi-stability and memory effects
Achievements in previous sections were used to design a genetic memory element. The closed chain neg(a,b) | neg(b,c)  | neg(c,d) | neg(d,a) demonstrating bi-stable characteristics was investigated in detail. The system arbitrarily started from expression (a,c) or (b,d) proteins. I investigated the circuit for different values of parameter η p but did not detect spontaneous transitions between states even at the extremely low η p = 0.00001. Flip-flops were detected at the constitutive transcription ε n = 0.01. Effective protein binding r > 1.0 improved stability, however intensive protein decay δ > 0.0001 destabilized the system (data not shown). Fortunately, the circuit demonstrated stable behavior at a standard range of parameters. After it dropped to an arbitrary state, the system survived for a long time (Figure 8a, b). However, the circuit was sensitive to external inputs. For example, when the system is in state bd, then a programmable input a can change the state to a new state ac, i.e. the production of b and d proteins can be changed to a and c proteins by input a (Figure 8c). More of them, if input a no longer exists, nevertheless, the system stayed at the state ac and did not turn back until the specific input b changed the system's state again (Figure 8d). The program code for this experiment is in the Appendix.

Bifan
Bifan is a simplest element of combinatorial logics based on multiple inputs. I investigated bifans with variable structures under the control of diverse inputs. These circuits represented a stable and relatively poor behavior at various parameters. Bifan comprising only pos gates performed synchronous outputs, when the input signals were applied separately (Figure 9a). Over time, when the inputs overlapped each other, the pos bifan worked as an adder (Figure 9b). Bifan consisting exclusively of neg gates behaved as an inverter or a subtractor depending on the time interval between inputs (Figure 9c). Symmetric bifan with two pos gates and two neg gates generated symmetric outputs (Figure 9d). Opposite, asymmetric bifans with three pos gates and one neg gate (Figure 9e) or  (Figure 9f) performed more complex jobs. These patterns were repeatable, recognizable, and strongly dependent on time and a combination of inputs even in the stochastic environment.

Discussion
I wondered how Kaufmann's networks could be combined with agent-based modeling to describe interactions between objects in terms of rules. One idea was to avoid a combinatorial explosion. The answer was found earlier by Robin Milner, Joachim Parrow, and David Walker who introduced π-calculus which are inherently modular. This concurrent language describes a dynamical network in terms of computable elements. Andrew Phillips and colleagues developed SPiM calculus that is in fact a language for biological systems. For example, gene is defined as a gate with corresponding inputs and outputs. Genetic networks are composed from simple computational elements. Modeling with Stochastic Pi-Machine is like programming, where program parts correspond to autonomous genetic elements. The design of a system from essential elements looks like a process of evolution. The "…compositional evolution is inherently scalable and leaves gradual evolution stuck at the starting blocks" (Watson, 2006). Therefore, the composition nature of π-calculus could allow a combinatorial design and perhaps genetic programming in the field.
A genetic constructor of five elements was used in this work; decay, null gate, gene product, negative and positive gates. Actually, the two last primitives are composed from the three first ones and only the gene product, as well as neg and pos gates were used to build more complex engines, such as genetic switches, oscillators, feedforward and feedback loops, pulse generators, memory elements and combinatorial logics based on multiple inputs -the essential blocks of large information devices and networks. This flexible toolkit allowed me to investigate genetic circuits of different topologies and complexities. An important part of the simulations was stochastic noise which is an internal property of biological systems that have a small number of molecules in a cell. The SPiM computing environment showed that some functions such as oscillations or stochastic pulses could be performed easily. For example, the SPiM Player used in the experiments is based on the Gillespite stochastic algorithm that enables accurate computer simula tions. Unfort una tely, t he al gor ith m is computationally too expensive for models with a large number of channels.
From another perspective, motifs in large-scale gene regulation networks were regarded like 'biological atoms' (Alon, 2007). They are negative and positive auto-regulators, coherent and incoherent feedforward loops, single-input module, multi-output feedforward loop, bifan and dense overlapping regulators. These network motifs discovered in nature by systems biology could possibly be used as building blocks for synthetic biology (Kuznetsov, 2009b).
In this paper, artificial and natural networks were investigated. At the beginning, I repeated the original models in SPiM calculus developed by (Blossey et al., 2006), such as inverter (Figure 1c), a chain of neg gates and the head feedback (Figure 2b-d). Then I made new designs, i.e. a robust genetic memory element ( Figure 8); genetic gates with two inputs and one output were introduced (Figure 5f, 6f), as well as a tristable toggle switch was described in terms of stochastic πcalculus (Figure 3d-f). The last example confirmed the possibility genetic networks with high connectivity -the most important feature of eukaryotes.
Different networks were compared, e.g. the repressilator comprising three neg genetic elements demonstrated more regular oscillations than the oscillation motif including two proteins (Figure 4c, d). Also, C1 coherent and I1 incoherent feedforward loops were investigated at various conditions. They were so stable that, for instance, C1-FFL behaved like OR logic element in a noise environment at the nominal parameters: r = 1.0, δ = 0.001, ε n = 0.1, η n = 0.01, ε p = 0.01, η p = 0.1 (Figure 5f). However, C1-FFL and I1-FFL produced spontaneous pulses with high amplitude at a very small ε p = 0.00001 (Figure 5d and Figure 6d).
The coherent and incoherent compositional FFL and FBL circuits generated coherent and decoherent pulses accordingly ( Figure 7). These circuits were analyzed systematically at various parameters, their diverse behaviors classified, and attention was given to state III which formed trains of pulses (S1: 15, S2: 15). Variations in parameters were more significant for pos gates than for neg ones. In particular, parameter η p playing a role of activator was critical (S2: [13][14][15][16]. Sometimes distinctions in a network topology and a range of parameters changed the system in a spectacularly unpredictable way. As such, the system demonstrated unusual behavior in state III during a long-time simulation that was impossible to see during a short-time run (S1: 15, S2: 15).
As mentioned, a memory element from four neg gates was constructed. The circuit follows two very stable states; nevertheless, an intensive external impulse can turn the system from one state to an alternative state. Operations can be repeated many times in both directions (Figure 8d). This model could explain epigenetic effects in living cells. Finally, bifans of various functionality formed particular outputs under controllable inputs. They showed a time-amplitude modulation with diverse patterns of expression ( Figure 9). The last design gives me hope that scalable systems with multiple inputs will be able to signal pattern recognition.
The stochastic π-calculus is inherently different from differential equations and allows us to investigate new phenomena. I was able to construct genetic switches and memory elements using stochastic dynamics and network topology variations and avoiding cooperative mechanisms (Figure 3 and Figure 8). My attempts to find memory effects in regulated positive and negative feedback loops were unsuccessful (Supplement 3). This is a discrepancy with the conclusion of Uri Alon (Alon, 2007) who pointed out a memory capability for these kinds of circuits. The disagreement prompted me to think of a more appropriative language for biological systems. It would be very interesting to know about the behavior of these circuits in real conditions in vivo and which approach is better. ues of parameters. The topology and complexity of networks played a significant role in their behavior. New networks emerged easily, sometimes because of duplications and transpositions of the previous ones. These operations resemble natural genetic mechanisms such as vertical and horizontal gene transfer -essential operators of biological evolution (Kuznetsov, 2007). Some variations in parameters are similar to 'point mutations' leading to an optimization (adaptation) of the system to a desirable pattern of behavior. Future investigations will shed light on which formalism -the stochastic πcalculus or Ordinary Differential Equations -is better able to describe biological entities.