Robustness analysis of biochemical network models

Biological systems that have been experimentally verified to be robust to significant changes in their environments require mathematical models that are themselves robust. In this context, a necessary condition for model robustness is that the model dynamics should not be sensitive to small variations in the model's parameters. Robustness analysis problems of this type have been extensively studied in the field of robust control theory and have been found to be very difficult to solve in general. The authors describe how some tools from robust control theory and nonlinear optimisation can be used to analyse the robustness of a recently proposed model of the molecular network underlying adenosine 3',5'-cyclic monophosphate (cAMP) oscillations observed in fields of chemotactic Dictyostelium cells. The network model, which consists of a system of seven coupled nonlinear differential equations, accurately reproduces the spontaneous oscillations in cAMP observed during the early development of D. discoideum. The analysis by the authors reveals, however, that very small variations in the model parameters can effectively destroy the required oscillatory dynamics. A biological interpretation of the analysis results is that correct functioning of a particular positive feedback loop in the proposed model is crucial to maintaining the required oscillatory dynamics.     

Translating biochemical network models between different kinetic formats

Mechanistic biochemical network models describe the dynamics of intracellular metabolite pools in terms of substance concentrations, stoichiometry and reaction kinetics. Data from stimulus response experiments are currently the most informative source for in-vivo parameter estimation in such models. However, only a part of the parameters of classical enzyme kinetic models can usually be estimated from typical stimulus response data. For this reason, several alternative kinetic formats using different “languages” (e.g. linear, power laws, linlog, generic and convenience) have been proposed to reduce the model complexity. The present contribution takes a rigorous “multi-lingual” approach to data evaluation by translating biochemical network models from one kinetic format into another. For this purpose, a new high-performance algorithm has been developed and tested. Starting with a given model, it replaces as many kinetic terms as possible by alternative expressions while still reproducing the experimental data. Application of the algorithm to a published model for Escherichia coli's sugar metabolism demonstrates the power of the new method. It is shown that model translation is a powerful tool to investigate the information content of stimulus response data and the predictive power of models. Moreover, the local and global approximation capabilities of the models are elucidated and some pitfalls of traditional single model approaches to data evaluation are revealed.     

Quantifying robustness and dissipation cost of yeast cell cycle network: the funneled energy landscape perspectives

We study the origin of robustness of yeast cell cycle cellular network through uncovering its underlying energy landscape. This is realized from the information of the steady-state probabilities by solving a discrete set of kinetic master equations for the network. We discovered that the potential landscape of yeast cell cycle network is funneled toward the global minimum, G1 state. The ratio of the energy gap between G1 and average versus roughness of the landscape termed as robustness ratio (RR) becomes a quantitative measure of the robustness and stability for the network. The funneled landscape is quite robust against random perturbations from the inherent wiring or connections of the network. There exists a global phase transition between the more sensitive response or less self-degradation phase leading to underlying funneled global landscape with large RR, and insensitive response or more self-degradation phase leading to shallower underlying landscape of the network with small RR. Furthermore, we show that the more robust landscape also leads to less dissipation cost of the network. Least dissipation and robust landscape might be a realization of Darwinian principle of natural selection at cellular network level. It may provide an optimal criterion for network wiring connections and design.     

Analysis of feedback loops and robustness in network evolution based on Boolean models

Background  

Many biological networks such as protein-protein interaction networks, signaling networks, and metabolic networks have topological characteristics of a scale-free degree distribution. Preferential attachment has been considered as the most plausible evolutionary growth model to explain this topological property. Although various studies have been undertaken to investigate the structural characteristics of a network obtained using this growth model, its dynamical characteristics have received relatively less attention.     

Consistent robustness analysis (CRA) identifies biologically relevant properties of regulatory network models

Background

A number of studies have previously demonstrated that “goodness of fit” is insufficient in reliably classifying the credibility of a biological model. Robustness and/or sensitivity analysis is commonly employed as a secondary method for evaluating the suitability of a particular model. The results of such analyses invariably depend on the particular parameter set tested, yet many parameter values for biological models are uncertain.

Results

Here, we propose a novel robustness analysis that aims to determine the “common robustness” of the model with multiple, biologically plausible parameter sets, rather than the local robustness for a particular parameter set. Our method is applied to two published models of the Arabidopsis circadian clock (the one-loop [1] and two-loop [2] models). The results reinforce current findings suggesting the greater reliability of the two-loop model and pinpoint the crucial role of TOC1 in the circadian network.

Conclusions

Consistent Robustness Analysis can indicate both the relative plausibility of different models and also the critical components and processes controlling each model.     

Bayesian sequential inference for stochastic kinetic biochemical network models.

As postgenomic biology becomes more predictive, the ability to infer rate parameters of genetic and biochemical networks will become increasingly important. In this paper, we explore the Bayesian estimation of stochastic kinetic rate constants governing dynamic models of intracellular processes. The underlying model is replaced by a diffusion approximation where a noise term represents intrinsic stochastic behavior and the model is identified using discrete-time (and often incomplete) data that is subject to measurement error. Sequential MCMC methods are then used to sample the model parameters on-line in several data-poor contexts. The methodology is illustrated by applying it to the estimation of parameters in a simple prokaryotic auto-regulatory gene network.     

Quantifying loopy network architectures

Biology presents many examples of planar distribution and structural networks having dense sets of closed loops. An archetype of this form of network organization is the vasculature of dicotyledonous leaves, which showcases a hierarchically-nested architecture containing closed loops at many different levels. Although a number of approaches have been proposed to measure aspects of the structure of such networks, a robust metric to quantify their hierarchical organization is still lacking. We present an algorithmic framework, the hierarchical loop decomposition, that allows mapping loopy networks to binary trees, preserving in the connectivity of the trees the architecture of the original graph. We apply this framework to investigate computer generated graphs, such as artificial models and optimal distribution networks, as well as natural graphs extracted from digitized images of dicotyledonous leaves and vasculature of rat cerebral neocortex. We calculate various metrics based on the asymmetry, the cumulative size distribution and the Strahler bifurcation ratios of the corresponding trees and discuss the relationship of these quantities to the architectural organization of the original graphs. This algorithmic framework decouples the geometric information (exact location of edges and nodes) from the metric topology (connectivity and edge weight) and it ultimately allows us to perform a quantitative statistical comparison between predictions of theoretical models and naturally occurring loopy graphs.     

Inference of biochemical network models in S-system using multiobjective optimization approach

MOTIVATION: The inference of biochemical networks, such as gene regulatory networks, protein-protein interaction networks, and metabolic pathway networks, from time-course data is one of the main challenges in systems biology. The ultimate goal of inferred modeling is to obtain expressions that quantitatively understand every detail and principle of biological systems. To infer a realizable S-system structure, most articles have applied sums of magnitude of kinetic orders as a penalty term in the fitness evaluation. How to tune a penalty weight to yield a realizable model structure is the main issue for the inverse problem. No guideline has been published for tuning a suitable penalty weight to infer a suitable model structure of biochemical networks. RESULTS: We introduce an interactive inference algorithm to infer a realizable S-system structure for biochemical networks. The inference problem is formulated as a multiobjective optimization problem to minimize simultaneously the concentration error, slope error and interaction measure in order to find a suitable S-system model structure and its corresponding model parameters. The multiobjective optimization problem is solved by the epsilon-constraint method to minimize the interaction measure subject to the expectation constraints for the concentration and slope error criteria. The theorems serve to guarantee the minimum solution for the epsilon-constrained problem to achieve the minimum interaction network for the inference problem. The approach could avoid assigning a penalty weight for sums of magnitude of kinetic orders.     

A new measure of the robustness of biochemical networks

MOTIVATION: The robustness of a biochemical network is defined as the tolerance of variations in kinetic parameters with respect to the maintenance of steady state. Robustness also plays an important role in the fail-safe mechanism in the evolutionary process of biochemical networks. The purposes of this paper are to use the synergism and saturation system (S-system) representation to describe a biochemical network and to develop a robustness measure of a biochemical network subject to variations in kinetic parameters. Since most biochemical networks in nature operate close to the steady state, we consider only the robustness measurement of a biochemical network at the steady state. RESULTS: We show that the upper bound of the tolerated parameter variations is related to the system matrix of a biochemical network at the steady state. Using this upper bound, we can calculate the tolerance (robustness) of a biochemical network without testing many parametric perturbations. We find that a biochemical network with a large tolerance can also better attenuate the effects of variations in rate parameters and environments. Compensatory parameter variations and network redundancy are found to be important mechanisms for the robustness of biochemical networks. Finally, four biochemical networks, such as a cascaded biochemical network, the glycolytic-glycogenolytic pathway in a perfused rat liver, the tricarboxylic acid cycle in Dictyostelium discoideum and the cAMP oscillation network in bacterial chemotaxis, are used to illustrate the usefulness of the proposed robustness measure.     

S-system-based analysis of the robust properties common to many biochemical network models

Robustness is a key feature to characterize the adaptation of organisms to changes in their internal and external environments. A broad range of kinetic or dynamic models of biochemical systems have been developed. Robustness analyses are attractive for exploring some common properties of many biochemical models. To reveal such features, we transform different types of mathematical equations into a standard or intelligible formula and use the multiple parameter sensitivity (MPS) to identify some factors critically responsible for the total robustness to many perturbations. The MPS would be determined by the top quarter of the highly sensitive parameters rather than the single parameter with the maximum sensitivity. The MPS did not show any correlation to the network size. The MPS is closely related to the standard deviation of the sensitivity profile. A decrease in the standard deviation enhanced the total robustness, which shows the hallmark of distributed robustness that many factors (pathways) involve the total robustness.     

The robustness of neutral models of geographical variation

The approximation of diploid migration by gametic dispersion is studied. The monoecious, diploid population is subdivided into panmictic colonies that exchange migrants. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to a new allele at the same rate u. Diploid-migration models without self-fertilization and with selfing at the “random” rate (equal to the reciprocal of the deme size in each deme) are investigated; in the gametic-dispersion models, selfing occurs at the random rate. It is shown for the unbounded stepping-stone model in one and two dimensions, the circular stepping-stone model, and the island model that the probabilitities of identity in state at equilibrium for diploid migration are close to those for gametic dispersion if the mutation rate is small or the deme size is large. Explicit error bounds are presented in all the above cases. It is also proved that if the number of demes is finite and the migration matrix is arbitrary but time independent and ergodic, then in the strong-migration approximation the equilibrium and the ultimate rate and pattern of convergence of both diploid-dispersion models are close to the corresponding gametic-dispersion formulae. For the strong-migration approximation at equilibrium, migration must dominate both mutation and random drift; for the convergence results, it suffices that migration dominate random drift. All the results apply to a dioecious population if the migration pattern and mutation rate are sex independent.     

Quasi-multiparameter sensitivity measure for robustness analysis of complex biochemical networks

Robustness is the ability to resume reliable operation in the face of different types of perturbations. Analysis of how network structure achieves robustness enables one to understand and design cellular systems. It is typically true that all parameters simultaneously differ from their nominal values in vivo, but there have been few intelligible measures to estimate the robustness of a system's function to the uncertainty of all parameters.We propose a numerical and fast measure of a robust property to the uncertainty of all kinetic parameters, named quasi-multiparameter sensitivity (QMPS), which is defined as the sum of the squared magnitudes of single-parameter sensitivities. Despite its plain idea, it has hardly been employed in analysis of biological models. While QMPS is theoretically derived as a linear model, QMPS can be consistent with the expected variance simulated by the widely used Monte Carlo method in nonlinear biological models, when relatively small perturbations are given. To demonstrate the feasibility of QMPS, it is employed for numerical comparison to analyze the mechanism of how specific regulations generate robustness in typical biological models.QMPS characterizes the robustness much faster than the Monte Carlo method, thereby enabling the extensive search of a large parameter space to perform the numerical comparison between alternative or competing models. It provides a theoretical or quantitative insight to an understanding of how specific network structures are related to robustness. In circadian oscillators, a negative feedback loop with multiple phosphorylations is demonstrated to play a critical role in generating robust cycles to the uncertainty of multiple parameters.     

Discriminating between rival biochemical network models: three approaches to optimal experiment design

Background  

The success of molecular systems biology hinges on the ability to use computational models to design predictive experiments, and ultimately unravel underlying biological mechanisms. A problem commonly encountered in the computational modelling of biological networks is that alternative, structurally different models of similar complexity fit a set of experimental data equally well. In this case, more than one molecular mechanism can explain available data. In order to rule out the incorrect mechanisms, one needs to invalidate incorrect models. At this point, new experiments maximizing the difference between the measured values of alternative models should be proposed and conducted. Such experiments should be optimally designed to produce data that are most likely to invalidate incorrect model structures.     

Quantifying the robustness of first arrival dates as a measure of avian migratory phenology

As the climate changes, many long‐term studies have shown that the timing of bird migration is shifting, increasing the need for reliable measures of migratory phenology. Ideally, daily counts of birds at a site are used to calculate the mean arrival date (MAD) but, as this approach is not always possible and is very labour‐intensive, simpler metrics such as first arrival date (FAD) have commonly been used. Here, we examine the relationship between FAD and MAD in 28 summer migrant bird species over a 42‐year period (1970–2011) at Portland Bird Observatory, UK. Although significant correlations between FAD and MAD were detected, relationships were weak, particularly in long‐distance migrants. We suggest that FAD, although a simple and straightforward measure, is not particularly robust as a proxy for overall migratory phenology at a population level.     

Construction of biochemical computer models

This paper described methods for constructing (digital) computer models of biochemical systems when the main object is to investigate the system itself, and not to fit experimental data (e.g., from tracer kinetics) to a set of equations. The author describes model-building as an art which is difficult to communicate, but nevertheless gives valuable tips on the conceptual and practical aspects, from his own considerable experience.     

A structured approach for the engineering of biochemical network models, illustrated for signalling pathways

Quantitative models of biochemical networks (signal transduction cascades, metabolic pathways, gene regulatory circuits) are a central component of modern systems biology. Building and managing these complex models is a major challenge that can benefit from the application of formal methods adopted from theoretical computing science. Here we provide a general introduction to the field of formal modelling, which emphasizes the intuitive biochemical basis of the modelling process, but is also accessible for an audience with a background in computing science and/or model engineering. We show how signal transduction cascades can be modelled in a modular fashion, using both a qualitative approach--qualitative Petri nets, and quantitative approaches--continuous Petri nets and ordinary differential equations (ODEs). We review the major elementary building blocks of a cellular signalling model, discuss which critical design decisions have to be made during model building, and present a number of novel computational tools that can help to explore alternative modular models in an easy and intuitive manner. These tools, which are based on Petri net theory, offer convenient ways of composing hierarchical ODE models, and permit a qualitative analysis of their behaviour. We illustrate the central concepts using signal transduction as our main example. The ultimate aim is to introduce a general approach that provides the foundations for a structured formal engineering of large-scale models of biochemical networks.