PMC:1570349 / 1409-7031 JSONTXT

{"project":"2_test","denotations":[{"id":"16859561-11955179-1688444","span":{"begin":1431,"end":1432},"obj":"11955179"},{"id":"16859561-12648679-1688444","span":{"begin":1431,"end":1432},"obj":"12648679"},{"id":"16859561-14766974-1688444","span":{"begin":1431,"end":1432},"obj":"14766974"},{"id":"16859561-17337276-1688444","span":{"begin":1431,"end":1432},"obj":"17337276"},{"id":"16859561-12447975-1688445","span":{"begin":1662,"end":1663},"obj":"12447975"},{"id":"16859561-15446975-1688446","span":{"begin":1664,"end":1665},"obj":"15446975"},{"id":"16859561-16081475-1688447","span":{"begin":1847,"end":1849},"obj":"16081475"},{"id":"16859561-11570588-1688448","span":{"begin":3489,"end":3491},"obj":"11570588"}],"text":"1 Background\nThe use of mathematical models provides tools for the analysis of complex molecular interactions aiming at an understanding of processes occurring in living cells. For many problems in cellular control, stochastic effects and time-delays can be ignored and systems of first-order ordinary differential equations (ODEs) can adequately model the underlying processes. Denoting by x and p the biochemical concentrations and parameters, respectively, the instantaneous change in x is described by the vector field f:\n= f(x, p).     (1)\nIn the study of such systems, an important goal is to understand how the observed physiological behavior arises out of gene network topology and parameters p. Some of these questions may be studied via examining the bifurcation manifolds Σ of the ODE system, which partition the parameter space into regions of different qualitative behavior (see e.g., [1] for a general overview to bifurcation theory). From ODE models and measured parameters, the forward problem of computing the bifurcation diagram has contributed significantly towards elucidating the complex mechanisms underlying cellular processes. For instance, mathematical and symbolic bifurcation analysis has led to an understanding of the possible dynamical behaviors that may arise out of simple gene systems (for a monograph, see [2], examples of more recent papers dealing with natural, designed, and model systems are [3-7]). For cell cycle models, bifurcation diagrams have given biologists a systems-level perspective of the roles played by the various constituent modules, as well as providing the ability to predict the behavior of mutant cells [8,9]. The desire to locate regions in parameter space exhibiting interesting dynamics has led to the development of computational tools for the automatic discovery of bifurcation sets [10].\nIn contrast, inverse problems have only recently attracted attention in biology as a way to unravel the workings of cellular mechanisms. In inverse problems one looks for causes for observed or desired effects [11]. Mathematically, such problems are typically ill-posed, in particular unstable; special mathematical techniques, called \"regularization methods\", have to be used to cope with this ill-posedness. Many variational and iterative regularization techniques have been developed over the years and applied to a variety of problems in science, engineering and finance (see [11] and some references quoted there, and [12]).\nIn the current context of cell biology, one would like to address problems such as: which parameter configurations lead to an observed qualitative behaviour of the system (\"identification\")? How can one introduce a certain qualitative behaviour into the system via parameter variations (\"design\")? We summarize such problems under the name of inverse bifurcation problems, where the task is to map the space of bifurcation diagrams back to the space of parameters. In particular, we consider inverse bifurcation problems of two types: identification and design. For the former, one would like to infer parameter values from observed bifurcation diagrams and hence the issue of uniqueness is typically of concern. For the latter, one is interested in parameter values that produce the desired outcome, hence uniqueness is not an issue. The notion of \"inverse bifurcation\" was first introduced into biology in [13]; more recently, another inverse bifurcation problem from mathematical ecology was studied in [14] by integral equation methods. These papers are concerned with the existence and uniqueness of nonlinear terms in equations realizing prescribed bifurcation diagrams and use analytical methods. In contrast, given the size and complexity of gene networks, we take a computational approach in addressing problems of inverse bifurcation. In this paper, we consider problems of moderate size for which the ill-posedness is not yet a crucial issue. Since ill-posedness increases with dimension, a major issue for larger problems will be to use appropriate regularization techniques.\nFor the design type of inverse bifurcation problems, there exists previous work in the engineering literature: The distance to bifurcation manifolds has been introduced to quantify the \"parametric robustness\" of system designs in [15]; in the context of design of chemical processes, optimization problems with constraints involving distance to bifurcation manifolds have been treated in [16]. For a recent review see [17]. For biological applications as we have them in mind, this issue of parametric robustness is also important. In addition, other geometric properties of the bifurcation diagram are of interest. These include the size of the parameter region resulting in bistability of solutions and the parametric distance between regions of different qualitative behavior. We will develop methodolgies by which inverse bifurcation problems involving the optimization of such quantities can be solved.\nIn this paper, we show the applicability of our inverse bifurcation algorithms to low dimensional gene systems. We formulate the inverse problems as constrained optimization problems, whose objective function and constraints involve geometric properties of bifurcation diagrams. We demonstrate that these problems can be solved efficiently by applying gradient-based nonlinear optimization algorithms in combination with one-parameter continuation methods to locate bifurcation points. The latter is a standard capability provided by existing bifurcation analysis software (see [1] for references to state-of-the-art numerical implementations)."}