Stochastic and deterministic modelling.

Stochastic and deterministic modelling.

1. Purposes of stochastic modelling

Deterministic modelling assumes the systems to be continuous and evolve deterministically. The behaviour of the system can be described using ODEs, which are then solved. However, such models ignore the phenomena that occur due to the fact that each system consists of a finite number of discrete particles, such as random fluctuations. For systems with very small particle numbers the deterministic models are not even appropriate because the concentrations are not continuous.

Stochastic modelling takes into account the fact that each system is composed of a finite and countable number of particles and considers the number of those particles similar to the way the deterministic system considers concentrations.

2. Drawbacks of stochastic modelling

2.1. Limits on particle numbers

Considering the fact that the number of particles in the system is very large, computational modelling of a stochastic method is very demanding and developing an algorithm is a complex task.

2.2. Lack of analysis methods

Stochastic modelling does not have such rigorously developed analysis methods as metabolic control analysis for deterministic modelling.

3. Drawbacks of deterministic modelling

3.1. Systems with small particle numbers

Stochastic methods consider random fluctuations which lead to significant change in system behaviour when the number of particles is small. Species are allowed to become extinct. In deterministic models the fluctuations are not accounted for and species concentrations never fall to zero. Therefore, in linear processes, the deterministic model behaviour will only be determined by difference in concentrations. The stochastic models can behave differently. This remains true even if stochastic systems have the same marginal distribution of system states.

3.2. Bi-Stable systems

Under deterministic simulation the system which is bi-stable will converge to the same stable steady state if the initial concentrations remain the same. Under stochastic simulation the system will converge to one of the two stable states, and it can not be predicted to which one. The probability of the system converging to each state, however, can be calculated.

4. Difference between the deterministic solution and the mean of stochastic solutions

It should be noted that if we repeat the stochastic simulation many times and calculate the mean, we will not end up with the same solution as the deterministic. This is only true for linear systems, but the solutions for nonlinear systems can be totally different.

5. Conclusion

Stochastic modelling should definitely be chosen when the particle numbers are in range where the concept of continuous concentration is no longer applicable or when the stochastic phenomena are themselves the object of research. The limit on the application of stochastic model is generally enforced at a certain particle number where computation becomes not feasible.

References

Pahle J, Biochemical simulations: stochastic, approximate stochastic and hybrid approaches, Briefings in Bioinformatics 2009, 10(1), pp 53-64

by Evgeny. Also posted on my website

Stoichiometry matrix

1. Definition

Stoichiometry matrix (SM) is a systematic arrangement of stoichiometric information from the reactions comprising the model. In a system with m species and n reactions the dimensions of the matrix are mxn. Chemical species are represented by rows and reactions – by columns. The elements of the matrix are corresponding stoichiometric coefficients. The selection of the system boundaries defines the complexity of the SM. When the concentration of a specie is considered fixed, the reaction is removed from the matrix.

The set of equations represented in the matrix together expresses the dynamics of the metabolite concentrations as

dS/dt = N*v,

where N is the matrix, v is the vector of fluxes and S is the vector of metabolite concentrations.

2. Applications

SM implies a steady state assuming that at any given time the concentration of the specie is constant. By using SM it is possible to enumerate all possible steady state flux solutions of a given network.

Personally, I like the fact that the SM is a crossroads of mathematics and biology, equally making sense for a person with a background in biology or information technology or mathematics.

2.1. Network reconstruction

The whole table of reactions encoded in the genome may be represented as SM. If the genes that encode for enzymes and reactions that each enzyme carries out are listed, the resulting table can be converted into the SM.

2.2. Mass conservation analysis

SM contains all information about the reaction network, therefore all necessary data to analyse mass conservation. Such relations can be retrieved from the SM as linear combinations of other rows. The result of removing all rows that are linear combinations of other rows is the reduced matrix which is used by software packages such as COPASI.

2.3. Stoichiometric modelling

In stoichiometric modelling, there are three major approaches are metabolic flux analysis (MFA), flux balance analysis (FBA) and metabolic pathway analysis (MPA). All three work by defining a high-dimension solution space of possible metabolic flux distributions based on the SM specifying system conservation relationships. The difference between the three approaches lies in how metabolic flux distributions are selected from the solution space.

MFA is a traditional approach which relies on extensive experimental data and computes a metabolic flux vector for a particular condition. Experimental data is used to simplify the SM.

FBA identifies only one optimal solution while alternative optimal solutions may exist. It very much depends on the validity of the model.

MPA, unlike the other two methods, can identify all metabolic flux vectors in a network. A finite set of solutions is achieved by additional constraints on the flux space.

References

Smolke C, The Metabolic Pathway Engineering Handbook: Fundamentals, CRC Press, 2009

Trinh T, Wlaschin A, Srienc F, Elementary Mode Analysis: A Useful Metabolic Pathway Analysis Tool for Characterizing Cellular Metabolism, Appl Microbiol Biotechnol, 2009, 81(5), pp 813-826

Wang X, Chen J, Quinn P, Reprogramming Microbial Metabolic Pathways, Springer, 2012

by Evgeny. Also posted on my website

COPASI and CellDesigner: Comparison

COPASI and CellDesigner are software packages for modelling and simulation of biochemical networks.

1. Common features

Models in both packages are based on compartments, species and reactions. Both packages have an area where the model tree is displayed, the area for entering and editing model parameters

2. Differences

2.1 CellDesigner

Networks are drawn based on the process diagram and are stored using SBML. CellDesigner supports a rich set of graphical elements for various compartments, species and reactions. There are predefines shapes for certain species, such as ‘truncated protein’ or ‘simple molecule’, or reactions, such as ‘transport’ or ‘dissociation’. Elements are added to the module by selecting them from a toolbar. Visual representation of the model is the strong advantage of CellDesigner. Figure 1 represents a model of MAPK-42 opened with CellDesigner. CellDesigner allows entering kinetic reactions by hand, but does not have predefined rate laws so they also have to be entered by hand for each reaction.

Figure 1 - CellDesigner with MAPK-42 model

2.2 COPASI

COPASI is less rich visually, but has advanced functionality for model simulation and analysis. The species are added to the model automatically by typing the attributes into the line in the list. Reactions can also be added directly by typing the chemical equation into the cell in the table. Unlike CellDesigner, COPASI has a number of predefined kinetic laws, can switch between stochastic and deterministic simulation methods and supports a number of methodologies for model analysis – computation of steady states, stoichiometric matrix, parameter estimation and optimisation. An important feature is the mathematical view where the model is described as a set of differential equations. Figure 2 represents the same model of MAPK-42 opened with COPASI, where differential equations are shown.

Figure 2 - COPASI with MAPK-42 model

3. SBML support

Both packages can read and write SBML, which is a file format for exchanging systems biology models. CellDesigner stores visual layout information in SBML annotations, but also has an option of exporting into ‘pure’ SBML in case conversion is required to a different software package which does not support annotations. CellDesigner can read and write SBML, and store rich enhanced functions as SBML annotations. COPASI converts its native model structure to SBML on exporting. Notes and annotations are preserved while importing and re-exporting as long as the objects that contains them is not deleted from the model.

4. Conclusion

While the strongest advantage of CellDesigner is the support for model building and editing via rich graphical representation, COPASI appears to be more suitable for simulation and analysis.

by Evgeny. Also posted on my website

Calculation of Phase Space Models with COPASI

COPASI is an open source modelling and simulation package for biochemical networks, capable of simulation based on ODEs and stochastic kinetics, and models can include discrete events.

A phase space diagram is a type of plots used to describe models that have complex dynamics. The axes represent the variables of the system, and each point in this space represents a state of the system. A trajectory in the phase-space is a line of points that describe the evolution of the state of the system, however time is not explicitly represented in such plots (even though it is implicit).

If the model has two variables, plotting them against each other is an obvious task. What if the model has three variables, like a model of calcium oscillations? At this time, COPASI can only display 2D plots. One solution is to display 2D projections of the phase space. For a model with 3 variables there are 3 possible projections: a vs b, a vs c and b vs c.

To create a phase space model, first download a model. In this case it is described in SBML. COPASI has an option for importing SBML: File -> Import SBML. COPASI may generate some warnings. In the left panel one can see the 3 species and 8 reactions included in the model.

To create a phase-space diagram locate Output Specifications -> Plots down in the left panel. In the Plots panel on the right click new, and a plot with a default name is created.

Double-click the plot in the panel. Now you can give it a meaningful name, and then click New curve. This is where you select the axis of the plot. In this diagram, transient concentrations of species will be plotted. Select a for X axis and b for Y axis and click OK.

Now a 2D curve appears in the right panel. Parameters can be edited to make them more intuitively understandable. Then, click Commit and select Tasks -> Time Course in the left panel. In the Time Course panel, click Run.

The result is the phase space diagram of the transient concentrations of species a vs b. Other plots are created in a similar way.

References

COPASI website
Kummer et al. - Model of calcium oscillations of Kummer et al. by Evgeny. Also posted on my website