Stochastic modelling of biochemical systems of multi-step reactions using a simplified two-variable model
- Qianqian Wu^{1},
- Kate Smith-Miles^{1},
- Tianshou Zhou^{2} and
- Tianhai Tian^{1}Email author
https://doi.org/10.1186/1752-0509-7-S4-S14
© Wu et al.; licensee BioMed Central Ltd. 2013
Published: 23 October 2013
Abstract
Background
A fundamental issue in systems biology is how to design simplified mathematical models for describing the dynamics of complex biochemical reaction systems. Among them, a key question is how to use simplified reactions to describe the chemical events of multi-step reactions that are ubiquitous in biochemistry and biophysics. To address this issue, a widely used approach in literature is to use one-step reaction to represent the multi-step chemical events. In recent years, a number of modelling methods have been designed to improve the accuracy of the one-step reaction method, including the use of reactions with time delay. However, our recent research results suggested that there are still deviations between the dynamics of delayed reactions and that of the multi-step reactions. Therefore, more sophisticated modelling methods are needed to accurately describe the complex biological systems in an efficient way.
Results
This work designs a two-variable model to simplify chemical events of multi-step reactions. In addition to the total molecule number of a species, we first introduce a new concept regarding the location of molecules in the multi-step reactions, which is the second variable to represent the system dynamics. Then we propose a simulation algorithm to compute the probability for the firing of the last step reaction in the multi-step events. This probability function is evaluated using a deterministic model of ordinary differential equations and a stochastic model in the framework of the stochastic simulation algorithm. The efficiency of the proposed two-variable model is demonstrated by the realization of mRNA degradation process based on the experimentally measured data.
Conclusions
Numerical results suggest that the proposed new two-variable model produces predictions that match the multi-step chemical reactions very well. The successful realization of the mRNA degradation dynamics indicates that the proposed method is a promising approach to reduce the complexity of biological systems.
Background
The advances in systems biology have raised the importance of quantitative methods for studying various systems in molecular biology. In recent years, various research methods, including mathematical modeling, statistical analysis, computer simulation and visualization, have been employed to investigate the dynamic or statistical properties of regulatory networks. In particular, mathematical models have been widely used to describe the dynamics of complex systems inside the cell, including genetic regulatory networks, cell signalling transduction pathways and metabolic pathways [1][2]. However, these substantial progresses have further raised a number of fundamental and challenging issues that require to be addressed imperatively.
One of the major challenges in systems biology is how to use simple mathematical models to describe complex biological systems. To address this issue, a number of modelling techniques have been designed. Among them, a widely used approach is to use one-step reaction to represent multi-step reactions, which is also called slow reaction. This technique is very important because recent theoretical and experimental studies have shown that a wide variety of biochemical events involve multi-step reactions [3]. Perhaps the most important example of multi-step reactions is transcriptional and translational processes that produce mRNA transcripts and proteins, respectively. Other examples include molecules (e.g. mRNA and protein) degradation and telomere length shortening processes. In fact, the process of multi-step reactions also exists in other areas such as organic chemistry and biophysical chemistry [4][5]. Therefore the major aim of this research work is to design simplified models to accurately characterize biological systems with multi-step reactions.
A widely used approach to simplify multi-step chemical reactions in the literature is to use one-step reaction. For example, the degradation process of mRNA or protein has been modelled by a first order reaction. However, since the one-step reaction cannot provide consistent description of the multi-step reactions, chemical reactions with time delay have been designed recently to model the multi-step chemical events more accurately [6][7][8][9]. Another important factor is noise in biological networks that may influence the system dynamics substantially. The deterministic modelling methods, which approximate molecular numbers using continuous concentrations [10][11], may not be appropriate to describe systems that contain species with small population numbers. To model stochastic systems more accurately, there are a few other ways. For example, we can use discrete Markov processes where the density of states of a well-stirred chemical reaction system at each time point can be represented by the chemical master equation (CME) [12][13]. One of the most well-known methods is called the stochastic simulation algorithm (SSA), which is a statistically exact method for simulating trajectories of the CME as the system evolves in time [14].
Furthermore, to deal with the intrinsic noise in reactions with time delay, the delay stochastic simulation algorithm (DSSA) was designed by introducing time delay into the SSA [15][16]. Unlike the SSA, which assumes that biochemical reactions are instantaneous and independent, the DSSA characterizes chemical systems that contain both fast and slow reactions. This delayed modelling approach has been applied to many physical and biological systems [16]. The DSSA was also extended to describe chemical events that have multiple delays or stochastic delay that follows a given probabilistic distribution [17][18]. In recent years, the DSSA has been widely used to simulate the dynamics of genetic regulatory networks and cell signalling pathways [7][19][20][21][22]. In addition, a number of effective simulation methods have been proposed to reduce the huge computing load of the DSSA [23][24][25][26]. Recently the work done by Luis Mier-y-Terán-Romero et al. opened some new aspects for the application of time delays in biological systems. Time delay may not be a constant that was assumed before [27]. Other modelling techniques proposed recently include the slow-scale linear noise approximation and stochastic quasi-steady-state assumption [28][29]. Most recently a new modelling approach has been proposed to simulate chemical reaction systems with memory reactions [30].
The degradation process of mRNA molecules is an important step in the regulation of gene expression, which also represents a typical system with multi-step reactions [31]. Although the mechanisms of mRNA degradation have been studied extensively during the last ten years, there are still a number of open problems with respect to the function of enzymes, structure of pathways and role of P-bodies, etc. in the regulation of mRNA degradation [32][33][34]. A major step in the quantitative study of mRNA degradation was the development of mathematical models based on the detailed chemical processes. A linear multi-component model was designed to investigate the nonsense-mediated decay of mRNA molecules in yeast [35][36]. This deterministic model for mRNA degradation process consists of 23 first-order reactions that describe transcription, translocation, ploy(A) shortening, decapping and digestion process. Computer simulations suggested that the widely used concept of half-life underestimated the averaged life-span of mRNA molecules; however, it is still a major factor that determines the life-span of different steps in the degradation pathway. In addition, robustness analysis showed that the change of degradation rate constant led to large variations of mRNA copy numbers. To interpret the complexity of mRNA degradation in a simpler manner, we proposed a multi-step reaction model using a chain of 11 chemical reactions, which gave very good approximation to the detailed one [37].
Chemical reactions with time delay has been used to further simplify mathematical models of mRNA degradation. Here time delay represents the time required in the multi-step reactions except the first reaction [37]. This simplified model was also extended to using stochastic time delay. However, numerical results showed that these first-order reaction models with delay did not give good approximation to the detailed degradation process [37]. Instead of using time delay to represent the missing intermediate reactions in the multi-step reaction, we recently proposed a new modelling approach by introducing a novel concept, namely the length of a molecule indicating its location in the multi-step reactions. Deterministic models using ordinary differential equations have been used to find the optimal value in a non-linear probability function [38]. However, it is still a challenge to apply this concept to stochastic models that are much more important than deterministic models for chemical reaction systems. Thus this work further validates the proposed model using stochastic simulations. We first introduce a new stochastic modelling method with two variables for describing chemical events with multi-step reactions, and then propose a stochastic simulation algorithm to numerically calculate the probability of the firing of the last reaction in the multi-step events. The efficiency and accuracy of the proposed method are examined by studying the mRNA degradation process of gene PRL30 based on experimental data.
Results and discussion
A new two-variable model
where B_{ i } are molecular species and k_{ i } rate constants. It is assumed that each molecule in the system will eventually turn to the product P or degrade if P = (). During this process, each molecule will pass through a number of states B_{1}, B_{2}, . . . , B_{ n } via the multi-step reactions.
In this work we use reactions (4) and (5) to design the two-variable reaction model.
Determination of probability function
Using this probability function, we designed an algorithm, namely Algorithm II in the Method section, to simulate the two-variable reaction model based on the SSA.
mRNA decay dynamics: case study for gene RPL30
to simulate the decay dynamics [40].
To model mRNA degradtion, Cao and Parker proposed a multi-component model that includes mRNA transcript synthesis, mRNA translocation, poly(A)-shortening process, and terminal deadenylation [35]. We have proposed a simplified model by putting a number of terminal deadenylation reactions into a single one [37]. This simple model is a typical multi-step reaction process. In this model, mRNA transcript is synthesized by a zero-order reaction S_{1}, then mRNA molecules translocate from the nucleus to cytosal via reaction S_{2}. The mRNA molecules in the cytosol produce proteins by the translational process, and in the meantime, the length of mRNA begins to decrease via a number of poly(A)-shortening reactions S_{3}, . . . , S_{9}. The final reaction in this process is the further exonucleolytic degradation S_{10}, which is regarded as the degradation reaction in this work, since the fragment product (FG) has no function to produce protein molecules.
Reactions and kinetic rates of the simplified stochastic model.
Reaction | Rate constant s_{ i } | Comment | |
---|---|---|---|
S _{1} | DNA → A | s _{1} | transcription |
S _{2} | A → B | s _{2} | transport |
S _{3} | B → BC 1 | s _{3} | full-length 70A-60A |
S _{4} | BC 1 → BC 2 | s _{4} | full-length 60A-50A |
S _{5} | BC 2 → BC 3 | s _{5} | full-length 50A-40A |
S _{6} | BC 3 → BC 4 | s _{6} | full-length 40A-30A |
S _{7} | BC 4 → BC 5 | s _{7} | full-length 30A-20A |
S _{8} | BC 5 → BC 6 | s _{8} | full-length 20A-10A |
S _{9} | BC 6 → BC 7 | s _{9} | full-length 10A-0A |
S _{10} | BC 7 → FG | s _{10} | fragment production |
Estimated parameters for the stochastic model of RPL30 and ACT1 mRNA degradations (Ratio = L_{0}/n X).
ACT1 construct | RPL30 construct | |||||
---|---|---|---|---|---|---|
X _{0} | Rate k | L 0 | Ratio | Rate k | L 0 | Ratio |
m = 5 | 0.1150 | 19 | 0.4222 | 0.1710 | 24 | 0.5333 |
m = 10 | 0.1110 | 37 | 0.4111 | 0.1660 | 47 | 0.5222 |
m = 20 | 0.1130 | 75 | 0.4167 | 0.1680 | 95 | 0.5278 |
m = 30 | 0.1130 | 112 | 0.4148 | 0.1670 | 142 | 0.5259 |
m = 40 | 0.1120 | 149 | 0.4139 | 0.1680 | 190 | 0.5278 |
m = 50 | 0.1120 | 186 | 0.4133 | 0.1670 | 237 | 0.5267 |
Using the estimated model parameters of the case X_{0} = 100, simulation results for the two constructs in Figure 5 (A) and (B) show that the two-variable model provides more accurate description of the mRNA degradation dynamics than the one-step model, in particularly for that in the first 25 minutes. For the ACT1 construct in Figure 5 (A), the optimal length number with ratio 0.412 gave more accurate simulation than the averaged length number. However, in Figure 5 (B) for the RPL30 transcript, the difference between the simulations using two different length numbers is small. In this case, the optimal ratio is 0.525, which is very close to 0.5.
To further examine the accuracy of the two-variable model, we used the stochastic model to simulate the mRNA dynamics using different initial transcript numbers. For each initial mRNA number, we generated 10, 000 simulations and then calculated the averaged mRNA numbers of all stochastic simulations. For both constructs in Figure 5 (C) and (D), our results show that there is small difference between the simulations using X_{0} = 5 and X_{0} = 10. However, there is not any significant difference between simulations when the initial mRNA number is larger than 10.
Conclusions
This work represents an attempt to use simplified mathematical models to describe complex biological systems. Concentrating on the chemical events of multi-step reactions, we proposed a new concept (e.g. the length of a molecule) as an additional measure to characterize system dynamics. The length of a molecule is defined as the location of a molecule in the multi-step reactions. Using the total molecule number and total length of molecules, we proposed a two-variable model to reduce the complexity of the multi-step reactions. The major contribution of this work is to design a nonlinear function that represents the probability of the firing of the last reaction in the multi-step reactions. To calibrate this probability function, we proposed a stochastic simulation method to calculate the probabilities of various system states. Numerical results suggested that this probability is dependent on the number of reaction steps but independent of the total molecule number, which suggested that we were able to design a simplified model based on the network structure. Then our proposed two-variable model was applied to simulate the dynamics of mRNA degradation using experimentally observed data. Numerical results suggested that the length of molecules, which is approximately a half of the maximal length initially, played an important role in realizing experimental data. The potential future work includes the application of the two-variable model to other multi-step reaction systems such as gene expression and telomere length regulation. In addition, the refinement of the two-variable model, such as the accuracy of the probability function, would also be very interesting.
Methods
Simulation algorithm for the probability function
The major part of this algorithm is to find frequency of the event B_{ n } = j based on the given length L and total molecule number X, which is explained as the following algorithm I.
- 1.
Set the total number of molecule X , number of reactions n, and initial full length L_{0} = nX.
- 2.Based on the following 10, 000 Monte-Carlo simulations, calculate the frequency freq(B_{ n } = j|L, X) for X molecules with total length L having j molecules, where j = 0, 1, . . . X and each of the molecule with length 1:
- (a)
Consider X molecules with full length. Denote the length of the i-th molecule as l_{ i } with i = 0, 1, . . . X.
- (b)
Use a random number r ~ U (0, 1) to select one molecule, with index j. If the length of that molecule l_{ j } > 1, reduce its length by 1, namely l_{ j } = l_{ j } − 1; if l_{ j } = 1, then repeat this step until finding a molecule with length greater than 1.
- (c)
Repeat step (b) for (L_{0} − L) times to get a set of molecules with total length L.
- (d)
Count the number of molecules in this set with length 1, denote as i, then update
- (e)
Repeat steps (a) ~ (d) for 10,000 times.
$freq\left({B}_{n}=i|L,X\right)=freq\left({B}_{n}=i|L,X\right)+1.$ - (a)
- 3.The probability for the last reaction firing is obtained by$f\left(X,L,n\right)={\displaystyle \sum _{j=1}^{X}}\frac{freq\left({B}_{n}=j|L,X\right)}{10000}\times \frac{j}{X}.$
Ordinary differential equation model
For a given initial condition B_{ i }(0), we obtained the analytical solution of the detailed system (16) and then solved the two-variable model (18) numerically using a stiff ODE solver ode 23s in MATLAB. We tested the solution of the two-variable model with different values of q based on different system conditions ranging from n = 5, 10, 15 as well as X = [5 10 50 100 200 500]. For each system condition, we selected the optimal value of q with which the two-variable model (18) generates simulation that is very close to that of the detailed ODE model (16). Finally we find the relationship between the value of q and system condition (X, L, n) by using a regression method [38].
An algorithm for simulating systems including two-variable model
The SSA is a general framework for simulating biochemical reaction systems. Now we propose an algorithm to incorporate the two-variable model into the SSA. It is assumed that a chemical reaction system is a well-stirred mixture at constant temperature in a fixed volume Ω. This mixture consists of N molecular species {S_{1}, . . . , S_{ N }} that chemically interact through M reaction channels {R_{1}, . . . , R_{ M }}. The dynamic state of this syetem is denoted as x ≡ (x_{1}(t), . . . , x_{ N } (t))^{⊤}, where x_{ i }(t) is the molecular number of species S_{ i } at time t. For each reaction channel R_{ j } (j = 1, . . . , M), a propensity function a_{ j } (x) is defined by a given state x(t) = x and the value of a_{ j } (x)dt represents the probability that one reaction will occur somewhere during the infinitesimal time interval [t, t + dt) [14][26][41]. In addition, a state change vector ν_{ j } is defined to characterise the change of molecular numbers due to the reaction R_{ j }. The element ν_{ ij } of ν_{ j } represents the change of the copy number of species S_{ i }. The algorithm for simulating chemical reaction systems with two-variable model is given below.
- 1.Calculate the values of propensity function a_{ j } (x) based on the system state x at time t. In particular, for the two-variable reaction with the total molecule number X (2) and total length L (3), the propensity function is a_{ j } = kX, where k is the harmonic mean of the rate constants (1), given by$k=\frac{n}{\frac{1}{{k}_{1}}+\cdots +\frac{1}{{k}_{n}}}.$(19)
- 2.Generate a sample r_{1} of the uniformly distributed random variable U(0, 1), namely r_{1} ~ U (0, 1), and determine the time of next reaction$\mu =\frac{1}{{a}_{0}\left(x\right)}\mathit{\text{ln}}\frac{1}{{r}_{1}}.$
- 3.Generate another sample r_{2} of U(0, 1) to determine the index k of the next reaction occurring in [t, t + µ],$\sum _{j=1}^{k-1}}{a}_{j}\left(x\right)<{r}_{2}{a}_{0}\left(x\right)\le {\displaystyle \sum _{j=1}^{k}}{a}_{j}\left(x\right)$
- 4.If the k-th reaction is not a two-variable model, update the state of the system by$x\left(t+\mu \right)=x\left(t\right)+{\nu}_{k}$
- 5.
Go back to step 1 if t + µ ≤ T, where T is the end time point. Otherwise, the system state at T is x(T) = x(t).
Declarations
Acknowledgements
This work is in part supported by grants from the Australian Research Council (ARC) (DP1094181 and DP120104460). T.T. is also in receipt of an ARC Future Fellowship (FT100100748). T.Z. is in part supported by grants 91230204 and 30973980 from the Natural Scientific Foundation of China.
Declarations
The publication costs for this article were funded by the corresponding author.
This article has been published as part of BMC Systems Biology Volume 7 Supplement 4, 2013: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2012: Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/7/S4.
Authors’ Affiliations
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