Topological augmentation to infer hidden processes in biological systems.

MOTIVATION: A common problem in understanding a biochemical system is to infer its correct structure or topology. This topology consists of all relevant state variables-usually molecules and their interactions. Here we present a method called topological augmentation to infer this structure in a statistically rigorous and systematic way from prior knowledge and experimental data. RESULTS: Topological augmentation starts from a simple model that is unable to explain the experimental data and augments its topology by adding new terms that capture the experimental behavior. This process is guided by representing the uncertainty in the model topology through stochastic differential equations whose trajectories contain information about missing model parts. We first apply this semiautomatic procedure to a pharmacokinetic model. This example illustrates that a global sampling of the parameter space is critical for inferring a correct model structure. We also use our method to improve our understanding of glutamine transport in yeast. This analysis shows that transport dynamics is determined by glutamine permeases with two different kinds of kinetics. Topological augmentation can not only be applied to biochemical systems, but also to any system that can be described by ordinary differential equations. AVAILABILITY AND IMPLEMENTATION: Matlab code and examples are available at: http://www.csb.ethz.ch/tools/index

Automatic generation of predictive dynamic models reveals nuclear phosphorylation as the key Msn2 control mechanism.

Predictive dynamical models are critical for the analysis of complex biological systems. However, methods to systematically develop and discriminate among systems biology models are still lacking. We describe a computational method that incorporates all hypothetical mechanisms about the architecture of a biological system into a single model and automatically generates a set of simpler models compatible with observational data. As a proof of principle, we analyzed the dynamic control of the transcription factor Msn2 in Saccharomyces cerevisiae, specifically the short-term mechanisms mediating the cells' recovery after release from starvation stress. Our method determined that 12 of 192 possible models were compatible with available Msn2 localization data. Iterations between model predictions and rationally designed phosphoproteomics and imaging experiments identified a single-circuit topology with a relative probability of 99% among the 192 models. Model analysis revealed that the coupling of dynamic phenomena in Msn2 phosphorylation and transport could lead to efficient stress response signaling by establishing a rate-of-change sensor. Similar principles could apply to mammalian stress response pathways. Systematic construction of dynamic models may yield detailed insight into nonobvious molecular mechanisms.

Efficient characterization of high-dimensional parameter spaces for systems biology.

BACKGROUND: A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling. RESULTS: We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another. CONCLUSIONS: Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.

Soliton fractals in the Korteweg-de Vries equation.

We have studied the process of creation of solitons and generation of fractal structures in the Korteweg-de Vries (KdV) equation when the relation between the nonlinearity and dispersion is abruptly changed. We observed that when this relation is changed nonadiabatically the solitary waves present in the system lose their stability and split up into ones that are stable for the set of parameters. When this process is successively repeated the trajectories of the solitary waves create a fractal treelike structure where each branch bifurcates into others. This structure is formed until the iteration where two solitary waves overlap just before the breakup. By means of a method based on the inverse scattering transformation, we have obtained analytical results that predict and control the number, amplitude, and velocity of the solitary waves that arise in the system after every change in the relation between the dispersion and the nonlinearity. This complete analytical information allows us to define a recursive L system which coincides with the treelike structure, governed by KdV, until the stage when the solitons start to overlap and is used to calculate the Hausdorff dimension and the multifractal properties of the set formed by the segments defined by each of the two "brothers" solitons before every breakup.

Sine-Gordon ratchets with general periodic, additive, and parametric driving forces.

We study the soliton ratchets in the damped sine-Gordon equation with periodic nonsinusoidal, additive, and parametric driving forces. By means of symmetry analysis of this system we show that the net motion of the kink is not possible if the frequencies of both forces satisfy a certain relationship. Using a collective coordinate theory with two degrees of freedom, we show that the ratchet motion of kinks appears as a consequence of a resonance between the oscillations of the momentum and the width of the kink. We show that the equations of motion that fulfill these collective coordinates follow from the corresponding symmetry properties of the original systems. As a further application of the collective coordinate technique we obtain another relationship between the frequencies of the parametric and additive drivers that suppresses the ratchetlike motion of the kink. We check all these results by means of numerical simulations of the original system and the numerical solutions of the collective coordinate equations.

Formation, control, and dynamics of N localized structures in the Peyrard-Bishop model.

We explore in detail the creation of stable localized structures in the form of localized energy distributions that arise from general initial conditions in the Peyrard-Bishop (PB) model. By means of a method based on the inverse scattering transform we study the solutions of PB model equations obtained in the form of planar waves whose amplitudes are described by the nonlinear Schrödinger equation (NLS). For localized initial conditions different from the pure N-soliton shape, we have obtained analytical results that predict and control the number, amplitude, and velocity of the NLS solitary waves. To verify the validity of these results we have carried out numerical simulations of the PB model with the use of realistic values of parameters and the initial conditions in the form of planar waves whose modulated amplitudes are given by the examples studied in the NLS. In the simulations we have found that N localized structures arise in agreement with the prediction of the analytical results obtained in the NLS.

Ratchet effect in a damped sine-Gordon system with additive and parametric ac driving forces.

We study in detail the damped sine-Gordon equation, driven by two ac forces (one is added as a parametric perturbation and the other one in an additive way), as an example of soliton ratchets. By means of a collective coordinate approach we derive an analytical expression for the average velocity of the soliton, which allows us to show that this mechanism of transport requires certain relationships both between the frequencies and between the initial phases of the two ac forces. The control of the velocity by the damping coefficient and parameters of the ac forces is also presented and discussed. All these results are subsequently checked by means of simulations for the driven and damped sine-Gordon equation that we have studied.

Domain wall dynamics in expanding spaces.

We study the effects on the dynamics of kinks due to expansions and contractions of the space. We show that the propagation velocity of the kink can be adiabatically tuned through slow expansions and/or contractions, while its width is given as a function of the velocity. We also analyze the case of fast expansions and/or contractions, where we are no longer on the adiabatic regime. In this case the kink moves more slowly after an expansion-contraction cycle as a consequence of the loss of energy through radiation. All these effects are numerically studied in the nonlinear Klein-Gordon equations (both for the sine-Gordon and for the potential), and they are also studied within the framework of the collective coordinate evolution equations for the width and the center of mass of the kink. These collective coordinate evolution equations are obtained with a procedure that allows us to consider even the case of large expansions and/or contractions.