Models Of System Biology

Physico-Mathematical Systems and System Biology Vikram Prakash, VIT University Vellore, [email protected] Abstract Many systems in nature, including biological systems, have very complex dynamics which generate random-looking time series. Biological system dynamics is basically understood by physico-mathematical models which gives the basics of the atomic/molecular interaction in living nature. The moment of the object in universe either with respect to time or space is made under the physico- mathematics. To better understand a particular dynamical system, it is often of interest to determine whether the system is caused by static subsystem, deterministic subsystems stochastic subsystems, or all. Alternatively, one can use methods that measure the complexity in a particular system which seldom make assumptions about a particular system, such as assuming the presence of stationarity. Additionally, mathematical and computational modelling techniques can be used to test different hypothesis about the dynamics of biological systems. Keywords Static model, Deterministic model, stochastic model, System Biology. Introduction Systems biology is a new approach to study biological phenomena. The main idea is to lift the reductionable paradigm that drives the current research practice in biology to a global view of biological systems that can be investigated at different level of models. The new paradigm is hypothesis driven, iterative and global. The main focus is on the functions that these components have in the dynamic evolution of biological systems. More abstractly we can say that biologists are passing from production of knowledge to organization of knowledge so that structure (well reported in many public data bases) must be related to behavior (still largely not organized in suitable representations). Static model Much of how an object responds to any interaction with its surroundings depends on what has happened to it up to now. Static modeling is better termed as an organism’s interactome, static models tend to be broader and coarser in scope, often encompassing the entire interactome. Static modeling is less demanding from an experimental perspective, as just about any assay on a population of cells will prove informative. This modeling is best conceptualized as the computerized reconstruction of molecular anatomy. Static modeling is about determining which elements are present and how they are interconnected it is a basic prerequisite for any kind of systems biological analysis. As one example, Determining whether two bacteria can metabolize the same sets of compounds requires an enumeration of their functional modules, roughly corresponding to the evolutionarily conserved sub graphs in their respective static models. As another example, identifying proteins which are essential for cellular function can be greatly aided by knowledge of which proteins are central in static network models. In particular, static models are essential starting points for more complex dynamic modeling strategies. Tasks Associated with Static Modeling: 1. Determine desired network detail. The first step in static modeling is to determine the scope and detail of the network reconstruction. 2. Enumerate input data sources. The next step is to work backward to determine which input variable could potentially predict the desired network properties. 3. Network reconstruction. 4. Experimental confirmation. In the ideal scenario, the properties of the predicted network are then experimentally tested. 5. Network applications. Given an experimentally reliable static network, we can then proceed to further applications, such as comparisons of networks across species and conditions (network alignment) and network-guided experimental prioritization. Limitations of Static Models: Perhaps the most obvious limitation of a static model is that it is in fact static: it does not incorporate temporal, spatial, or conditional information except indirectly. In particular, less detailed static models may give little information about how different nodes talk to each other. For example, low-resolution models that predict solely whether two proteins “interact” with some probability are useful for generating hypotheses, but give little mechanistic insight as to whether they are related by physical contact, presence in the same pathway, or regulation of the same genes. These limitations can be partially overcome by including more types of datas, though there are fundamental limitations on the level of conditional detail possible in a static network. Dynamic Models (Deterministic and Stochastic) Discrete models require the states of the system variables (genes, proteins, signaling molecules) to take on integer values. Although at a molecular level this requirement is the most realistic, it is often used at a higher level to simplify the resulting models. A boolean model provides one such simplification: it consists of binary-valued variables whose interrelationships are captured by boolean functions. In cases where a boolean model is too coarse grained for a particular system, a more elaborate dynamic Bayesian network can be used. These models can be either discrete or continuous, and they allow dynamical systems to be described probabilistically. An example of a recent discrete DBN applied to yeast cell cycle time series data is found in Zou and Conzen (2005). Short of molecular dynamics simulations that track the simultaneous position and velocity of every molecule in the system, the most realistic (and mechanistic) signaling network models fall under the stochastic chemical kinetics framework (Gillespie 2007). These models represent biological systems as well-stirred collections of finite numbers of chemical species; reactions are simulated probabilistically according to known reaction propensities Continuous models permit system variables to take on non-negative real-valued states. We focus on so-called chemical kinetic (mechanistic) models where states represent concentrations of molecules. These models, though approximate, are sufficiently accurate when the molecular populations of all species are orders of magnitude larger than one (Gillespie 2007). The oldest and most common modeling formalism uses ordinary differential equations (ODEs) and known chemical kinetic/physico-chemical principles (Cornish-Bowden 1979) to deterministically model molecular concentrations as a function of time. Though these equations are not usually analytically solvable, there exist a wide variety of numerical tools that can efficiently model relatively complex systems (Rangamani and Jyengar 2007). Partial differential equation (PDE) models of signaling networks describe the evolution of molecular concentrations as functions of both space and time. These models are more physically realistic than ODEs, but they are also significantly more difficult to solve and typically require custom-made numerical solution methods (Eungdamrong and Iyengar 2004). The addition of a noise term to a deterministic differential equation yields a stochastic differential equation (SDE), which in chemical kinetic systems often takes the form of a chemical Langevin equation (CLE) (Gillespie 2000). The CLE follows from approximations to discrete stochastic chemical kinetics, and its solution can be computed much more efficiently than solutions for the corresponding discrete model (Wilkinson 2009). Discrete dynamical models represent an active area of research in systems biology, and they have recently been discussed elsewhere (Uhrmacher et al. 2005).The cellular environment is constantly changing as a result of deterministic chemical reactions and stochastic fluctuations. Thus, dynamical systems are more realistic depictions of biology than static models. Through simulation, dynamical models enable characterization of nonlinear, emergent behavior that evolves over time. Such behavior is often only visible at a systems level and would be missed by reductionist methods (Bhalla and Iyengar 1999).The outputs of differential equation models relate more closely to experimentally observed phenotypes than coarser-grained alternatives (Sauer 2004). As a result, though these models often require extensive parameterization, the parameter space can be constrained such that the model reproduces experimental data. This significantly reduces the complexity of model calibration and also enables easier model validation (Rangamani and Iyengar 2007). Tasks Associated with Dynamical Modeling 1. Model construction and calibration. The first step is to specify the structure and parameterization of a model from prior knowledge and experimental data. 2. Model validation and testing. After calibration, it is important to compare model output with existing experimental data (Eungdamrong and Iyengar 2004; Ideker et al. 2001). This procedure is necessary (though not sufficient) to determine whether a model is specified correctly. 3. Parameter sensitivity analysis. Sensitivity analysis involves determining which molecular concentrations or kinetic parameters have the greatest influence on model behavior. This is valuable when prioritizing parameters for subsequent experimental measurement or perturbation (Rangamani and Iyengar 2007). 4. Analysis of emergent behavior. As mentioned emergent behavior arises from systems level properties that are not apparent from studying individual components. Many of these phenomena, which can include robustness to noise, feedback, bistability, and oscillation, are best characterized through simulation of the model (Gilbert et al 2006; Angeli et al 2004). 5. Predictive modeling and discovery. One of the most exciting areas of systems biology is prospective modeling to test hypotheses that are too difficult or expensive to query in vivo. Here, a prerequisite for making accurate predictions is a sufficiently detailed and accurate model (You 2004). Ordinary Differential Equation Systems Ordinary differential equation models are by far the most common dynamical model used in biology (Andrews and Arkin 2006). They represent behavior at the level of chemical kinetics, whereby the concentration of each system component yi(t) as a function of time is represented in the following manner: where y (t) = _yi (t) , . . . , yn (t)_ and fi is a function which describes the rate of change of yi(t). This function can be constant (uninhibited synthesis), linear (first-order reaction such as degradation), or nonlinear (second-order reaction like Michaelis–Menten kinetics), and its precise form follows from qualitative prior experimental knowledge. These coupled expressions are often collectively referred to as reaction rate equations (RREs). The RREs of most biologically realistic systems cannot be solved analytically, but numerous welldeveloped and efficient numerical methods for solving these systems are available. Assumptions of ODE Biological Models: The relative ease with which ODE models of biological systems can be constructed and solved is a consequence of the simplifying assumptions made about the system. These assumptions include as follows: • Reactions occur in a homogeneous, well-stirred volume (corollary: molecular concentrations are functions of time and not space) • Reactions occur in a deterministic manner • Discrete effects on molecular concentrations can be ignored (corollary: molecular populations of all species are orders of magnitude larger than one) Partial Differential Equation Systems Biological systems are known to exhibit spatial inhomogeneity, and some tasks require explicit modeling of the spatial dimension. This is especially true when the biological system in question extends across several cellular organelles, each potentially containing different components, or when the diffusion of individual components across the modeled space cannot be treated as an instantaneous process. Compartmental ODE models have been successfully used to model the former case, where components are assumed to be well mixed within compartments and transport between compartments occurs at a much slower measurable rate (Aldridge et al. 2006a). As these models are modified versions of the ODE models described above, we will not discuss them further. In the latter case, i.e., when explicitly modeling the diffusion of certain components, partial differential equation models are necessary. Here, the spatial dimension is modeled as a continuous quantity, and the concentration of each component becomes a function of both space and time. The PDEs most commonly used to describe such systems are reaction–diffusion equations, where the concentration of each component yi(t) of the system can be represented as follows (derived using Fick’s second law of diffusion): Where y(t) is as above, Di is a diffusion coefficient, xj represents a spatial dimension, and m is the number of spatial dimensions modeled. The first term on the RHS, fi, describes the contributions of chemical reactions to the time derivative, and the second term describes the contributions of diffusion. Compared to ODE models, PDE systems are much more challenging to solve, in part because they require many more parameters (Eungdamrong and Iyengar 2004). Aside from the kinetic parameters needed to specify fi, the reaction–diffusion system requires a diffusion coefficient for each species (which are difficult to measure experimentally (Rangamani and Iyengar 2007)), and fluxes and/or concentrations of each component must be specified at the boundary of the physical space being modeled. This latter constraint becomes even more prohibitive when considering complex physical geometries. Solutions to nonlinear PDE systems are almost exclusively numerical, and the added realism of the model comes at a computational cost due to the increased dimensionality of the system. PDE Models Describing Biological Systems Models of biological systems governed by PDEs employ two of the three simplifying assumptions of ODE models, with spatial homogeneity being the exception. Nonetheless, when mathematically and computationally tractable, these models can accurately reproduce spatially varying molecular behavior. One of the first examples of a PDE model describing a biological system modeled the behavior of two (generic) morphogens reacting and diffusing through simple geometries of cells (Turing 1952). Subsequent work elaborated upon this simple model of morphogen-controlled patterning. One study simulated a four morphogen reaction–diffusion system to mimic pattern formation in Drosophila embryogenesis (Lacalli 1990). By adjusting model parameters, the authors could produce striped patterns of morphogen concentration compatible with observed wild-type and mutant phenotypes. Stochastic Differential Equation Systems Both ODE and PDE models of biological systems assume that reactions occur in a deterministic manner. This assumption seems to imply that biological reactions exhibit little to no heterogeneity or stochasticity (“intrinsic noise”), which is known to be false (McAdams and Arkin 1997). Rather, the main reason for the success of deterministic biological models is that stochastic effects are often rendered negligible by averaging across large numbers of molecules or cells. This phenomenon also underlies the success of continuous mechanistic models, where discrete numbers of molecules can be approximated with continuous concentrations. There are, however, a number of well-characterized biological systems where the modeling assumption of deterministic reactions leads to qualitatively incorrect depictions of behavior. A deterministic model of the circadian rhythm oscillator parameterized with particular degradation rates fails to oscillate; in contrast, the noise present in the corresponding stochastic model gives rise to more robust oscillatory behavior (Vilar et al. 2002). In a common class of biochemical reaction mechanisms, enzymatic futile cycles, extrinsic noise (i.e., noise due to components/ processes outside the system) in a stochastic model was shown to induce bi-stable oscillatory behavior that was absent in a similar deterministic model (Samoilov et al. 2005).These (and other) important exceptions to deterministic reaction mechanisms have led to the application of stochastic models to biological systems. We begin by representing the state of the system as a function of time with Z(t) = [Zi (t) , . . . , Zn (t)], where Zi(t) represents the number of molecules of species i. Capital letters are used to emphasize the stochastic nature of the model; the Zi’s are random variables. A specific instantiation of the system is represented by lowercase letters; i. e., z = [zi, . . . , zn]. The system state can be altered by the firing of any of p reactions; each reaction changes the state by vk = [v1 k, . . . , vnk], 1 ≤ k ≤ p, where vik represents the change in the number of molecules of species i after the completion of reaction k. Each reaction can be characterized by its propensity function ak (z), defined so that ak (z) dt is equivalent to the probability that reaction k will occur once in the system in the infinitesimal time interval [t, t + dt] given Z (t)=z. Given that any instantiation of the system is random, it would be useful to have a probabilistic expression for the time evolution of the system P (z, t|z0, t0) (probability that the system is in state z at time t, given that it is in state z0 at time t0). Using the above quantities and the laws of probability, this can be derived as follows: Equation (1.1) is called the chemical master equation (CME). Since the possible values of z are discretely varying, the CME is actually a set of coupled ODEs that is nearly as large as the number of possible combinations of molecules in the system. Consequently, except for very simple systems, these equations are not solvable analytically and numerical solutions are usually intractable. Progress has been made in developing approximation schemes for numerically solving the CME (Munsky and Khammash 2006; Deuflhard et al. 2007; Jahnke and Huisinga 2008), but most applications turn to Monte Carlo methods to sample from the distribution P (z, t|z0, t0). The stochastic simulation algorithm (SSA), also known as the Gillespie algorithm, simulates each reaction sequentially as they occur in time (Gillespie 1977). This approach has been widely used in stochastic modeling of biological networks, in part because it produces a draw from the exact probability distribution that solves the CME. Conclusion With thousands of sequenced genomes (Wheeler et al. 2007) and hundreds of functional genomic data sets (Barrett et al. 2005), the future of systems biology is bright. In static modeling, the supervised learning approach, in which high-throughput data is compared against a small training set of curated knowledge, has proven to be the most fruitful data integration strategy to date. In particular, supervised predictions of function and interaction from multiple data sets are more robust than those derived from individual data sets and have provided a foundation for recent work on network alignment and systematic validation. The primary challenges for static modeling are to (1) decide on a set of reference networks and (2) tie every predicted node and edge in such networks to a gold-standard experimental test such as co-immunoprecipitation for confirmation of physical protein interactions. These steps will be crucial to bringing network predictions to the same level of confidence and widespread utilization as gene predictions. For dynamic models, the core problem is that the area will remain data starved (Albeck et al. 2006) until high-throughput methods for the determination of rate constants (Famili et al. 2005) and spatial substructure (Foster et al. 2006; Schubert et al. 2006) become commonplace. Recent efforts at compiling and curating a number of biological constants (Milo et al. 2009) and developing a repository of systems biology models (Hucka et al. 2003) are an important step in the right direction toward establishing a repository of “consensus constants ” References 1. Bernie J. Daigle, Jr., Balaji S. Srinivasan, Jason A. Flannick, Antal F. 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