5 Modeling Process of Biological Systems
5.2 Mathematical Modeling: xHPN for Biological Applications
The creation of a model for a biological system generally leads to the following advantages (Dunn 2003)
1. Modeling improves understanding. The comparison of model predictions and biological behavior leads to an increased understanding of the considered processes. The results of simulations give some indication of the occurrence of observed phenomena that are inexplicable till now. Additionally, the model formulation itself improves the understanding because complex cause-effect sequences and interactions have to be translated into a mathematical formalism.
2. Modeling supports experimental design. Experiments have to be designed such that the model can be tested sufficiently. The model itself usually indicates which experimental data are needed to identify the model parameters. Sensitivity analysis can reveal that some parameters have negligible effects on the model and, thus, these effects can be neglected from the model and the experiments while other parameters have a deep impact on the model and, hence, the experiments have to focus on these processes.
3. Models used for predictions. Once a model is established and verified, it can be used to predict the behavior of the regarded system under different environmental conditions.
4. Models used for process optimization. A verified model can be used for optimizing processes which relate to profits or costs to enable, for example, an open-loop control of these processes.
Numerous model formalisms have been proposed for modeling and simulation of biological systems (see e.g. (Wiechert 2002)). Generally, there needs to be a distinction between qualitative and quantitative approaches. Qualitative models represent only the fundamental compounds, their interaction mechanisms, and the relationships amongst them while quantitative models describe, in addition, the time-related changes of the components.
Hence, a qualitative model is the basis for every quantitative model and the mentioned improved data basis enables us to extend qualitative models to quantitative ones today.
Beyond this, quantitative model formalisms can be further divided into discrete, continuous, and hybrid approaches as well as into deterministic and stochastic techniques.
Figure 5.3: Different types of cell representation (Chmiel and Briechle 2008, Dunn 2003) Furthermore, models of biological systems can be classified according to their complexity (Chmiel and Briechle 2008, Dunn 2003). Figure 5.3 gives an overview of the different perspectives to represent a population. On the one hand, models are classified according to the amount of components that are needed to represent the cellular system. If a model consists of several differentiable components, it is called structured; otherwise, if the system is represented by one component, it is called unstructured. Unstructured models disregard intercellular processes and describe the system only based on changes in its environment. On
unstructured structured
non-segregatedsegregated
Approximation by assuming balanced growth
Approximation by assuming an average individual
Reduction of the cell population to one component system
Cells consist of several components but the population
is reduced to one average cell
The population is regarded as heterogeneous system of differentiable individuals but the
cells are reduced to a one component system
The population is regarded as heterogeneous system of differentiable individuals and the
cells consist of several components
extreme simplification greatest possible idealization
minor simplification realistic situation
the other, models can be divided into segregated models which regard the system as a heterogeneous collection of differentiable individuals and non-segregated models which approximate the system by an average individual. Segregated models consider the heterogeneity of the population concerning cell age, size, growth rate, and physiological state to allow a more precise description of the system.
The decision which modeling approach to use is difficult and strongly influenced by the availability of data. If all kinetic data is known, models consisting of ordinary differential equations are mostly the first choice while in the absence of this kinetic data only qualitative approaches are usable. An additional difficulty arises in the demand of simultaneously having a model which is easy to understand and an abstraction of the real system as well as a detailed and nearly complete description of it. Besides, the modeling process of biological systems is further complicated by incomplete knowledge, noisy and inaccurate data, and different ways of representing data and knowledge.
Figure 5.4: Petri net extensions: From a basic Petri net to an extended hybrid Petri net for biological applications
Petri nets with their various extensions are a universal graphical modeling concept for representing biological systems in nearly all degrees of abstraction. They support both the qualitative and the quantitative modeling approach. Once a qualitative Petri net model has been established, the quantitative data can be added successively. The Petri nets in Section 4.1 and 4.2 are examples for qualitative models due to the fact that no time is associated with the transitions. The arcs can be provided with the stoichiometric of the respective reaction and the tokens represent a discrete quantity of species. The qualitative analysis of such models considers all possible behaviors of the system at any time. Timed, stochastic, continuous, and hybrid Petri nets are examples of quantitative models (see Section 4.3, 4.4, and 4.5). The time is associated with the Petri net behavior by assigning each transition a delay, a hazard, and a maximum speed, respectively. Furthermore, the biological processes can be modeled
discretely as well as continuously and, in addition, discrete and continuous processes can also be combined within one Petri net model to so-called hybrid Petri nets (see e.g. (David and Alla 2001)). The Petri net formalism with all its extensions is so powerful that all other formalisms are included and, hence, only one formalism is needed regardless of the approach (qualitative vs. quantitative, discrete vs. continuous, deterministic vs. stochastic) which is appropriate for the respective system. The Petri net formalism is easy to understand for all researchers from different disciplines (biology, mathematics, informatics, and system sciences) which work together in the modeling process and is an ideal way for intuitive representing and communicating experimental data and knowledge of biological systems.
Besides, Petri nets allow hierarchical structuring of models and offer the possibility of different detailed views for every observer of the model. For these reasons, the developed xHPN formalism is superior to systems of ordinary differential equations which are mostly the first choice for modeling biological systems.
The Petri net formalism developed and used in this work has been gained from many discussions with biologists and biotechnologist to satisfy all their requirements and to represent with it nearly all kinds of biological reactions and phenomena. The result is called extended Hybrid Petri Net - abbreviated as xHPN and depicted Figure 5.4 - and the precise definition is given in Definition 4.68 (Section 4.5). The abbreviation has been chosen in such a general manner to emphasize that this formalism is not only useable for biological processes but also for nearly all other processes e.g. production, business, or communication (see Chapter 8). This xHPN formalism is extended by providing each Petri net element with a biological meaning. This extension is called xHPNbio (extended Hybrid Petri Nets for biological applications) and the formal definition is given below.
Definition 5.1 (xHPNbio)
An xHPNbio is an xHPN (see Definition 4.68) with a concrete transformation of xHPN elements to biological ones. This transformation is summarized in the following table by mentioning also some examples of the biological meaning.
xHPN element Biological meaning
Places
Biological compounds
metabolites, enzymes, substances, substrates, products, signals, genes, proteins, cells, complexes, activators, inhibitors, repressors, DNA, RNA
Transitions
Biological processes
biochemical reactions, metabolic reactions, interactions, regulatory reactions, signal transduction reactions, chemical reactions, binding, phosphorylation
Tokens/Marks Quantities of biological compounds molecules, concentrations, cells
Normal arcs Connections of biological compounds and processes Test arcs
Activation of biological processes
transcription process, activation in gene regulation, enzyme activity, activation mechanisms
Inhibitor arcs Inhibition of biological processes
repression of gene regulation, inhibition mechanisms Read arcs Needs for biological processes
catalysis
Arc weights Biological coefficients
stoichiometric coefficients, yield coefficients Min/max. capacities Reasonable biological capacities
biological knowledge
Delays Duration of biological processes
Hazard functions Random duration of biological processes stochastic kinetics
Maximum speeds Rate of biological processes kinetics effects/laws
xHPNbio
Biological systems
metabolic networks, signal transduction networks, regulatory networks, chemical networks, cell cycle, cell communication, diseases, population dynamics, flux networks, cultivation processes The following table gives some examples of biological reactions and their modeling with the xHPNbio formalism.
Table 5.1: Examples for modeling biological reactions with the xHPNbio formalism Type of
reaction Example Petri net
Production →
discrete or continuous
Degradation →
discrete or continuous
Chemical
reaction → 2
Chemical reaction modeled by mass action kinetics
3 → 2
⋅
Biochemical reaction modeled by Michaelis-Menten kinetics
→
⋅
loop-connection or read arc
Inhibition reaction
Activation reaction
Positive gene regulation
Negative gene regulation
S P
⋅ E
S
⋅
S P
E
S1
S2
P I
S1
S2
P A