BioNetS makes use of elementary reactions (zeroth, first and second order). The following examples illustrates each type of reaction:

In the above reactions, the calligraphic letters denote a single molecule of a chemical species. The number of molecules of a particular species in the system at time t is denoted with uppercase letters (e.g., A(t), B(t), A_B(t), and V(t)). All the rate constants, γ, δ, and k₁-k₆, have units of per time. Eq. 1 represents a process in which a molecule A is produced when the reaction proceeds in the forward direction and is degraded in the reverse direction. In the forward direction the reaction is zeroth order and proceeds with an average rate of γ. In the backward direction, the reaction is first order, and the average rate of degradation is δA(t). The forward reaction in Eq. 2 represents a process in which chemical species A is converted to species B. In this case A and B might represent two different conformations of the same molecule. In Eq. 2 both the forward and backward reactions are first order because the reaction rates are proportional to the respective concentrations. The forward reaction given in Eq. 3 is a second order reaction in which an A molecule and a B molecule come together to form the complex A_B. The average rate for the reaction is k₁A(t)B(t). The backward reaction is a first order reaction in which A_B dissociates at an average rate of k₂A_B(t). In Eq. 4 the forward reaction produces a molecule V. The difference between this reaction and the forward reaction in Eq. 1 is that the average rate is k₃V(t). This leads to exponential growth of V(t). This reaction is particularly useful if V(t) is interpreted as the cell volume. In the backward reaction, two V molecules come together and degrade one of the V molecules. The average rate for this reaction is k₄V(t)(V(t) - 1). The V(t) - 1 term arises because two of V(t) molecules must be chosen to react. This type of term also arises in reactions that produce homodimers. This reaction eventually stops the exponential growth of V. The net effect of these two reactions is to produce logistic growth. The total average reaction rate for the set of reactions given in Eqs. 1–4 is

where F_i and B_i are the average forward and backward rates, respectively, for the ith reaction.

For the rest of this section, we assume that the volume of the cell is not changing and only consider Eqs. 1–3. In the Examples we consider a case in which the volume is changing. If A(t), B(t) and A_B(t) are present in large numbers, then the law of mass action can be applied to derive equations that govern the concentrations [A]= A(t)/V, [B]= B(t)/V and [AB]= A_B(t)/V, where V is the cell volume. These equations are

The primed rate constants indicate that they have been appropriately scaled by the volume (i.e, k'₃= k₃V and γ' = γ/V), and, therefore, have units of either per time per concentration or concentration per time. Note that to convert to units of molar, we also have to appropriately scale the rate constants by Avagadro's number. Eqs. 6–8 represent a macroscopic description of the process, because they ignore fluctuations in the concentration that arise from the stochastic nature of chemical reactions.

In general, A(t), B(t) and A_B(t) are random variables that take on any nonnegative integer value. The Gillespie algorithm 32 can be used to generate sample paths of the process. This algorithm assumes that the random time ΔT_i, between the ith and i + 1 reaction, is exponentially distributed. For the simple example given by Eqs. 1–3, the mean waiting time between reactions, which characterizes the exponential distribution, is μ_ΔTi = γ + δ A(t_i) + k₁A(t_i) + k₂B(t_i) + k₃A(t_i)B(t_i) + k₄A_B(t_i), where t_i is the time at which the ith reaction occurred. Therefore, t_i⁺¹ = t_i + ΔT_i. Once the time at which the next reaction occurs has been determined, the following probabilities are used to determine which reaction occurred:

Once the reaction has been determined, the chemical species are updated accordingly. As discussed in the Numerical methods section, BioNetS uses an efficient implementation of the Gillespie algorithm 33.

Another description of discrete stochastic processes is achieved through use of the master equation that governs how the probabilities of the various random variables in the process evolve in time. Let p_{a, b,a_b}(t) = Pr [A(t) = a, B(t) = b, A_B(t) = a_b], then P_{a,b,a_b}(t) satisfies the master equation

The master equation is the starting point for deriving various approximate schemes for describing the system 28. In the next section, we discuss an approximate scheme that is valid in the limit of large, but finite molecule numbers. The simplest approximation scheme is achieved by considering the first moments of the process. We will use over bars to denote averaging. For example, . Eq. 15 can be used to derive equations that govern the time evolution of all the first moments. Because of the second order reaction in Eq. 3, the equations for the means are coupled to the second moments. In fact, the nth moment equations contain terms that involve the n+ l moments. Thus, there is no closure to the system. The simplest closure scheme is to assume that all moments factorize (e.g., ). This represents the macroscopic limit in which fluctuations are ignored. In this limit, we recover Eqs. 6–8 from the master equation.

The general form of the master equation for a system consisting of N chemical species and M reactions is

where n is a N-dimensional vector of species numbers, F_i and B_i are the backward and forward rates for the ith reaction, and the vectors δ_i contain the stoichiometric constants for the ith reaction. For the simple model given by Eqs. 1–3, N = 3, M = 3, and p_n(t) = Pr[A(t) = n₁, B(t) = n₂, and A_B(t) = n₃]. The forward and backward rates are F₁ = γ, B₁ = δn₁, F₂ = k₁n₁, B₂ = k₂n₂, F₃ = k₃n₁n₂, and B₃ = k₄n₃. The δ_i vectors are the rows of the stoichiometric matrix

The (i,j) element in the above matrix represents the change in the jth chemical species when the ith reaction proceeds in the forward direction.

If the molecule numbers are large as compared to 1, then the master equation Eq. 16 can be approximated by the continuous process 2835

where

This result can be derived in several ways. One method is to note that Eq. 15 represents a second order finite differencing of Eq. 18, with a grid size of 1. Another method is to make use of the shift operator

where f(n) is an arbitrary smooth function and for our purposes k is an integer. If the shift operator is used in Eq. 15, the diffusion limit is achieved when the Taylor series expansion given in Eq. 21 is truncated at j = 2.

Sample paths consistent with Eq. 18 can be generated using the following set of SDEs

where the w_k(t) are independent Gaussian white noise processes. These equations are often referred to as the chemical Langevin equations. For Eqs. 1 – 3, the explicit form of the SDEs are

BioNetS generates numerical solutions to the SDEs given by Eq. 22 using either an explicit or semi-implicit Euler method. The form of these methods is

where ε = 0 for the explicit method and ε = 1 for the semi-implicit method and the Z_k(t) are independent standard normal random variables. The advantage of using the chemical Langevin equations is that in the appropriate parameter regime, numerical solutions to the set of SDEs given by Eq. 22 can be generated much more efficiently than using the Gillespie algorithm. We expand upon this point in the Examples section. Higher order numerical algorithms for SDEs are available 36, but the noise structure of the chemical Langevin equations makes these schemes very cumbersome to implement. In the Examples, we verify that the Euler methods given by Eq. 26 are sufficient to produce reliable results. We note that the Δ matrix is generally sparse, and BioNetS takes advantage of this sparseness to optimize the efficiency of the two Euler methods (see Numerical Methods, below).

It is often desirable to allow some of the chemical species to be treated as continuous random variables and some to be treated discretely. This is particularly true for the case of transcriptional regulation by transcription factors. In this situation there can be as few as one DNA/transcription factor binding site and mRNA abundances can be as small as 10 or fewer. In contrast, protein abundances can be in the thousands. The technical difficulty with implementing hybrid schemes that include both discrete and continuous random variables is that the Gillespie method requires constant transition rates between reactions. This may not be the case, if some of the chemical species are evolving continuously in time. BioNetS overcomes this problem in one of two ways.

Let N_d <N be the number of discrete chemical species and M_d ≤ M the number of reactions that produce a change in one of the N_d chemical species. The overall reaction rate at time t_j for the discrete set of chemical species is

If the time step Δt for the SDEs is small enough such that

then p_t is approximately the probability of a transition in Δt. In the above equation ε is a user specified tolerance. The probability of two discrete transitions in Δt is proportional to (Δt)². Choosing ε < 0.1, which means the probability of two reactions in Δt is less than 0.01, generally produces good results. However, this should be verified on a case by case basis. At each time step, BioNetS checks to verify that Ineq. 28 is satisfied for the specified ε. If so, a uniform random number R is generated and compared against p_t. If R < p_t, then a transition occurred and the conditional probability R/p_t is used to determine which of the discrete transitions occurred. If p_t > ε, then the discrete reactions determine the fastest time scale in the system. In this case the Gillespie algorithm is used to update the discrete reactions, and the random time step Δt_j is used to update the SDEs.

A pseudo-code description of the above algorithm is displayed in Table 3:

We begin with a simple system that consists of the following two reactions:

In this system, monomer molecules M are produced at an average rate γ and degraded at an average rate δ_mM(t). Two monomers can then bind to form a dimer molecule D. The average forward and backward rates for the this reaction are k₁M(t)(M(t) - 1) and k₂D(t), respectively. The dimers are degraded at a rate δ_d. We will treat two cases. In the first case the cell volume is assumed to be constant, and in the second case the cell is allowed to grow and divide. To model cell growth, the cell volume V_c is treated as a random variable V_c = αV, where V is a non-negative discrete random variable and α represents a unit of volume. The random variable V is governed by the reaction

The above reaction causes V to grow exponentially fast with an average rate of k₃. Note that logistic growth is produced when the backward reaction in Eq. 32 is included.

We start by considering the simple case in which the volume of the cell remains constant. To use BioNetS follow these steps. Copy BioNetS onto your machine, and double click to launch. Help is included as part of the program, and accessed from the Help menu. The Help document will walk you through all the steps needed to enter reactions and run the simulator.

The user interface asks you to enter the reaction and corresponding rate constants in the top part of the script window as shown in Fig. 1a. In the bottom part of the script window, you can toggle between panels. The Species panel is shown in Fig. 1a and allows the user to specify how the simulator treats each chemical species, discrete or continuous. The Constants panel lists the order in which the rate constants are referenced. The Output panel allows the user to specify the ouput type. There are two ways to generate program output, either binary or ASCII. Binary output is based on MATLAB binary files, so it is possible to drive the program with MATLAB and use MATLAB's plotting routines to view the output. It is also possible to generate time series and histograms of the data from within BioNetS. Using ASCII files for I/O allows the simulator to be run through shell scripts. The Executable panel allows the user to generate either an executable file or source code. BioNetS generates portable C/C++ code that can be compiled and run in any computing environment. BioNetS can directly compile the C/C++ code. However, this requires the Developer tools, included on all recent Apple machines and available directly from http://developer.apple.com for free. The compiled code can then be run from within BioNetS. The Comments panel is available for the user to enter descriptive comments about the model.

To run BioNetS as a BioSpice agent, you need to move the source directory onto a OAA-supported system. Once there, open up the MakeOAA file and specify the locations of your oaalib folder. Then just type "make -f MakeOAA" (without the quotes) to create the agent.

Figures 2A and 2B show plots of time series for the monomer number generated by BioNetS. The parameter values used to generate these figures are given in the caption. Figure 2A is the result obtained when M and D are treated as discrete variables. Figure 2B is the result from the chemical Langevin equations. The solid line shown in both panels is the result from the following equations for the first moments:

Figure 3A shows the probability density function (PDF) of the dimer concentration at various times for the discrete and continuous case. Notice that for all times, the agreement between the two different methods is very good. At the final time, t = 200 s, the system has reached steady state. These figures indicate that the chemical Langevin equations are accurately capturing the dynamics and steady-state behavior of the discrete system.

In this section we describe how cell growth and division can be modeled using BioNetS. We will assume that the cell is experiencing exponential growth up until the time it divides. As discussed above, the cell volume V_c = αV is treated as a random variable. In this model cell division occurs when V exceeds a threshold value V_max. Note that the choice of V_max influences the degree of variability observed in the cell division times: cells growing from V = 1 to V_max = 2 will have a large amount of variability in their division times, while those growing from V = 100 to V_max = 200 will have less variable times, and those ranging from V = 1000 to V_max = 2000 will be still less variable. Changing the range of V in this way requires rescaling the relationship of V to the cell volume by adjusting the value of α. When cell division occurs the volume is halved, and the proteins are randomly divided between the two cells using a binomial distribution. Only one of the daughter cells is tracked. Because second order reactions require two molecules to collide, the rate constants for these reactions should scale like k₁ = k'₁/V_c. We also assume that the production rate of monomers scales as γ = γ'V_c. This is a reasonable assumption, because as the cell grows the transcription and translation machinery increases. These assumptions produce the following rate equations for the concentrations

The terms in Eqs. 35 and 36 that involve k₃ arise because of dilution due to cell growth. We use the same parameter values as in the constant volume case except δ_m = 1 and δ_d = 0. The cell growth rate is k₃ = 0.02 (assuming a scaling of 1 time unit to one minute, this yields an average cell division time of ln 2/k₃ ≃ 35 minutes, typical for bacteria), the scale factor for the cell volume is α = 1 (for simplicity), and V_max = 100. With these choices of parameter values, Eqs. 35 and 36 are identical with Eqs. 33 and 34, and we expect the average behavior of this system to be similar to that of the constant volume case.

The screen shot shown in Fig. 1B illustrates how this model is entered into BioNetS. Figure 4A shows the time series for the volume and monomer number treating each variable discretely. The results for the continuous case are virtually identical. In Fig. 4B the concentration is plotted as a function of time. This figure should be compared with Fig. 2A. The solid line in Fig. 4B is the result from solving Eqs. 35–37. Figure 3B shows the PDFs for the dimer concentration at various times. Both the discrete and continuous results are shown. By comparing Fig. 3A with Fig. 3B, we see not surprisingly that for this simple example the main effect of volume growth is to act as an additional noise source and increase the variability of the distributions.

We next use BioNetS to simulate a two gene system that has been studied in the literature 37. In this system, the protein A coded for by gene a acts as an activator for gene a and gene r, by binding to the promoter regions, P_a and P_r, of the respective gene. This increases the rate of mRNA_a and mRNA_r production by a factor α_aand α_r, respectively. The protein R, acts as a represser for both genes by binding to A to form the inactive complex A_R. All gene products, mRNA and protein, are actively degraded. However, the heterodimer A_R protects the R subunit from degradation. The system consists of 9 chemical species and the following 14 biochemical reactions:

Figure 5 shows the way this system is entered into BioNetS.

An interesting feature of the system is that it is capable of producing sustained oscillations 37. Figure 6A shows a times series for the repressor protein number when all the chemical species are treated as discrete random variables. The values of the rate constants used to generate this figure are listed in Fig. 5.

The chemical species P_a, P_r, P_r_A, and P_r_A are binary random variables: they can only take on the values 0 or 1. Therefore, these species can not be approximated as continuous random variables. All the other chemical species appear in sufficient quantities to justify the continuum approximation. Figure 6B shows a time series corresponding to Fig. 6A using the hybrid model. The hybrid model was run using the semi-implicit Euler method, and for these parameter values, runs 3 times faster than full model. Visually, the agreement between the two methods appears good. To test the accuracy of the Euler method, we used BioNetS to construct 2-D histograms of R versus mRN A_r. The results for the discrete and hybrid models are shown in Figs. 7B and 7B. To construct these histograms 10, 000 oscillations were used. Excellent agreement between the discrete and hybrid model is seen. This indicates that the hybrid model is accurately sampling the steady-state distribution. To verify that the hybrid model faithfully captures the dynamics of the system, we computed the power spectra of both models. The results are shown in Figs. 8A and 8B. Again, excellent agreement is seen between the discrete and hybrid model.

Using standard techniques in modern molecular biology, it is possible to design novel systems of promoter-gene pairs, such that virtually any desired regulatory network architecture may be instantiated; such networks are often called "synthetic gene networks." Recent implementations have included direct negative 22 and positive 23 feedback, a bistable switch 12, an oscillator 11, an intercellular communication system 38, and a bimodal self-activating system 39.

In this example, we use BioNetS to implement a model of a simple, open-loop network based around a novel engineered promoter, which has been designed and constructed by N. Guido and J. J. Collins at Boston University. The promoter, called O_RO_lac, combines the O_lac, O_R1, and O_R2 operator sites, so that it is repressed by the lac repressor protein (LacI) and activated by the lambda repressor protein (CI); see Fig. 9. Experiments have been conducted in which the promoter, along with other sites to produce the activator and repressor proteins, is integrated into a high copy number plasmid and inserted into a strain of Escherichia coli. The promoter's activity is observed using a fluorescent reporter, Green Fluorescent Protein (GFP). A detailed modeling study with direct comparisons to experimental results has been carried out using a fully discrete stochastic approach, and will be reported elsewhere (McMillen et al., manuscript in preparation). Our goal here is to provide a reasonably complex test case to evaluate the performance of BioNetS.

The processes to be captured by the model are: transcription and degradation of mRNA strands; translation of mRNA into protein; degradation of protein; formation of protein multimers (dimers in the case of CI, tetramers in the case of LacI); LacI binding to isopropyl-β-D-thiogalactopyranoside (IPTG), a chemical inducer that reduces LacI's binding affinity for O_lac; and protein-DNA binding at the O_RO_lac promoter's operator sites. We define the following chemical species: G, GFP; M_g, mRNA coding for GFP; X, CI monomer; X₂, CI dimer; M_x, mRNA coding for CI; D_x, the arabinose-inducible pBAD promoter site producing CI; Y, LacI monomer; Y₂, LacI dimer; Y₄, LacI tetramer; I₀, IPTG (present in massive excess and thus taken to be constant); Y_I, LacI tetramer bound to IPTG; M_y, mRNA coding for LacI; and D_y, the P_LtetO1 site constitutively producing LacI. In addition to these, we define species D₀ through D₈, representing the various permutations of proteins bound to the three operator sites in the O_RO_lac promoter (see Table for a list). There are twelve combinatorial possibilities, but we eliminate three of them on the basis that CI (X₂) binding O_R2 but notO_R1 is unlikely, because of the low binding affinitity of CI for O_R2 compared toO_R1. Table also lists the effect on the basal rate of production when the promoter is in each state. This reflects the regulatory effect of the proteins; for example, CI bound to O_R2 leads to a 10-fold increase in transcription rate, while LacI bound to O_lac halts transcription completely (note that we assume in the event of simultaneous binding of activator and repressor, repression "wins" and transcription is halted).

The following irreversible reactions represent the processes of transcription, translation, and degradation:

As in previous reactions, the caligraphic letters represent individual molecules of each species. We scale all times and rates by the cell division time.

Experimental measurements generally provide equilibrium rather than rate constants, and thus when writing reversible reactions we use the following notational convention: a reaction with equilibrium constant K has forward rate constant KR and backward rate constant R, where R is a scaling factor which sets the speed at which the reaction approaches equilibrium (we will consider three values of R: 1, 10, and 100). Using this notation, we represent protein-protein binding with the following set of reactions

Finally, protein-DNA binding is given by:

In all, the system consists of 21 species, participating in 34 reactions. The reactions are entered into BioNetS using the same method described in the previous examples. We use BioNetS' ability to represent individual species as either discrete or continuous to formulate three versions of the model: fully discrete, fully continuous, and a hybrid version in which the DNA species D₀ through D₈ are discrete while all other species are continuous. We vary the value of R, the scaling factor for reversible reactions, and keep all other parameters fixed at the following nondimensionalized values: β_g = 0.1, β_y = 1, β_x = 0.5, β_T = 10, γ_mrna = 3.5, γ_prot = 0.7, K_y = 0.01, K_y2 = 0.1, K_yI = 2 × 10^-6, K_x = 0.05, K₁ = 0.3, K₂ = 2K₁,K₃ = 0.008, K₄ = 1.4 × l0^-4K₃, I₀ = 1 × 10⁶.

To evaluate the steady-state probability distributions produced by the reaction system, simulations 250000 cell cycles in length were used to accumulate histograms (a built-in feature of BioNetS) of the number of molecules of GFP (species G), for each of the three versions of the model. As Fig. 10A shows, the resulting distributions are essentially identical, indicating that the continuum approximations used in the fully continuous and hybrid forms of the model were valid. Not all species in the system are well approximated as continuous variables: Fig. 10B shows the continuous probability distribution for species D₈, representing a promoter fully populated with two CI dimers and a LacI tetramer-IPTG complex. This situation is very rare in our parameter regime, and the system spends essentially all of its time with D₈ = 0. The fully continuous model, however, fluctuates into negative values, indicating that the continuum approximation has broken down. This does not significantly affect the distribution for GFP because the other, more common DNA states dominate the system's behavior; note, however, that if we were considering genomic DNA rather than a high copy number plasmid, we would not be able to employ a fully continuous model. The hybrid model, by treating the DNA species as continuous, eliminates the fluctuations into negative values. In general, the appropriate approximations will depend on both the system and the variables of interest: in the present example, if we were interested in the behavior of the operator sites themselves, we would not be able to use the fully continuous version of the model, but as a model solely of GFP expression the approximation suffices. Comparisons between types of models should be made to test the underlying assumptions, and BioNetS facilitates this process.

We used simulations 200 cell cycles in length to test the speed at which the three model versions ran. In each case, 200 simulations were run using a consistent set of 200 different random seeds; all runs were started with identical initial conditions. For the fully continuous and hybrid systems, the semi-implicit scheme was numerically stable and yielded consistent histograms for all time step sizes between dt = 0.001 and dt = 0.5, but the latter corresponds to just two time points per cell division cycle (recall that all times are scaled by the cell division time), and we chose instead to sample 20 points per cycle and set dt = 0.05. As shown in Table 2, the fully continuous method was always fastest, with the degree of improvement over the exact, fully discrete method depending strongly on the value of R, the scaling factor for the reversible reaction rates. For R = 1, the fully continuous method was only 1.4-fold faster than the fully discrete method, but as R is increased this speed advantage increases to over 4-fold at R = 10, then to over 30-fold at R = 100. (Note that the speed advantage of the fully continuous over the fully discrete method increases with the abundances of the chemical species. Shifting parameters to generate higher protein numbers can yield cases in which the continuum approximation is hundreds of times faster than the discrete approach; runs not shown here.) Use of a hybrid discrete/continuous method did not, for this particular model system, offer any speed gain over the fully discrete approach; the increased time involved in computing the Jacobian for the semi-implicit method is more time-consuming than simply simulating the reactions directly. Optimizing efficiency requires testing various potential approaches, and BioNetS makes this a simple process.