MOTIVATION: Our recent work introduced a generic method to construct the design space of biochemical systems: a representation of the relationships between system parameters, environmental variables and phenotypic behavior. In design space, the qualitatively distinct phenotypes of a biochemical system can be identified, counted, analyzed and compared. Boundaries in design space indicate a transition between phenotypic behaviors and can be used to measure a system's tolerance to large changes in parameters. Moreover, the relative size and arrangement of such phenotypic regions can suggest or confirm global properties of the system. RESULTS: Our work here demonstrates that the construction and analysis of design space can be automated. We present a formal description of design space and a detailed explanation of its construction. We also extend the notion to include variable kinetic orders. We describe algorithms that automate common steps of design space construction and analysis, introduce new analyses that are made possible by such automation and discuss challenges of implementation and scaling. In the end, we demonstrate the techniques using software we have created. AVAILABILITY: The Design Space Toolbox for MATLAB is freely available at http://www.bme.ucdavis.edu/savageaulab/ CONTACT: [email protected]

Robustness of organisms is widely observed although difficult to precisely characterize. Performance can remain nearly constant within some neighborhood of the normal operating regime, leading to homeostasis, but then abruptly break down with pathological consequences beyond this neighborhood. Currently, there is no generic approach to identifying boundaries where local performance deteriorates abruptly, and this has hampered understanding of the molecular basis of biological robustness. Here we introduce a generic approach for characterizing boundaries between operational regimes based on the piecewise power-law representation of the system's components. This conceptual framework allows us to define "global tolerance" as the ratio between the normal value of a parameter and the value at such a boundary. We illustrate the utility of this concept for a class of moiety-transfer cycles, which is a widespread module in biology. Our results show a region of "best" local performance surrounded by "poor" regions; also, selection for improved local performance often pushes the operating values away from regime boundaries, thus increasing global tolerance. These predictions agree with experimental data from the reduced nicotinamide adenine dinucleotide phosphate (NADPH) redox cycle of human erythrocytes.

MOTIVATION: The method of mathematically controlled comparison has been used for some time to determine which of two alternative regulatory designs is better according to specific quantitative criteria for functional effectiveness. In some cases, the results obtained using this technique are general and independent of parameter values and the answers are clear-cut. In others, the result might be general, but the demonstration is difficult and numerical results with specific parameter values can help to clarify the situation. In either case, numerical results with specific parameter values can also provide an answer to the question of how much larger the values might be. In contrast, a more ambiguous result is obtained when either of the alternatives can have the larger value for a given systemic property, depending on the specific values of the parameters. In any case, introduction of specific values for the parameters reduces the generality of the results. Therefore, we have been motivated to develop and apply statistical methods that would permit the use of numerical values for the parameters and yet retain some of the generality that makes mathematically controlled comparison so attractive. RESULTS: We illustrate this new numerical method in a step-by-step application using a very simple didactic example. We also validate the results by comparison with the corresponding results obtained using the previously developed analytical method. The analytical approach is briefly present for reference purposes, since some of the same key concepts are needed to understand the numerical method and the results are needed for comparison. The numerical method confirms the qualitative differences between the systemic behavior of alternative designs obtained from the analytical method. In addition, the numerical method allows for quantification of the differences and it provides results that are general in a statistical sense. For example, the older analytical method showed that overall feedback inhibition in an unbranched pathway makes the system more robust whereas it decreases the stability margin of the steady state. The numerical method shows that the magnitudes of these differences are not comparable. The differences in stability margins (1-2% on average) are small when compared to the differences in robustness (50-100% on average). Furthermore, the numerical method shows that the system with overall feedback responds more quickly to change than the otherwise equivalent system without overall feedback. These results suggest reasons why overall feedback inhibition is such a prevalent regulatory pattern in unbranched biosynthetic pathways.

Abstract Motivation: When dealing with questions that concern a general class of models for biological networks, large numbers of distinct models within the class can be grouped into an ensemble that gives a statistical view of the properties for the general class. Comparing properties of different ensembles through the use of point measures (e.g. medians, standard deviations, correlation coefficients) can mask inhomogeneities in the correlations between properties. We are therefore motivated to develop strategies that allow these inhomogeneities to be more easily detected. Results: Methods are described for constructing ensembles of models within the context of a Mathematically Controlled Comparison. A Density of Ratios Plot for a given systemic property is then defined as follows: the \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(y\) \end{document}axis represents the value of the systemic property in a reference model divided by the value in the alternative model, and the \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(x\) \end{document}axis represents the value of the systemic property in the reference model. Techniques involving moving quantiles are introduced to generate secondary plots in which correlations and inhomogeneities in correlations are more easily detected. Several examples that illustrate the advantages of these techniques are presented and discussed. Contact: [email protected] * To whom correspondence should be addressed.

MOTIVATION: Mathematical models are the only realistic method for representing the integrated dynamic behavior of complex biochemical networks. However, it is difficult to obtain a consistent set of values for the parameters that characterize such a model. Even when a set of parameter values exists, the accuracy of the individual values is questionable. Therefore, we were motivated to explore statistical techniques for analyzing the properties of a given model when knowledge of the actual parameter values is lacking. RESULTS: The graphical and statistical methods presented in the previous paper are applied here to simple unbranched biosynthetic pathways subject to control by feedback inhibition. We represent these pathways within a canonical nonlinear formalism that provides a regular structure that is convenient for randomly sampling the parameter space. After constructing a large ensemble of randomly generated sets of parameter values, the structural and behavioral properties of the model with these parameter sets are examined statistically and classified. The results of our analysis demonstrate that certain properties of these systems are strongly correlated, thereby revealing aspects of organization that are highly probable independent of selection. Finally, we show how specification of a given behavior affects the distribution of acceptable parameter values.