Splitting the dynamics of large biochemical interaction networks

This article is inscribed in the general motivation of understanding the dynamics on biochemical networks including metabolic and genetic interactions. Our approach is continuous modeling by differential equations. We address the problem of the huge size of those systems. We present a mathematical tool for reducing the size of the model, master-slave synchronization, and fit it to the biochemical context.     

Selfish cellular networks and the evolution of complex organisms

Human gametogenesis takes years and involves many cellular divisions, particularly in males. Consequently, gametogenesis provides the opportunity to acquire multiple de novo mutations. A significant portion of these is likely to impact the cellular networks linking genes, proteins, RNA and metabolites, which constitute the functional units of cells. A wealth of literature shows that these individual cellular networks are complex, robust and evolvable. To some extent, they are able to monitor their own performance, and display sufficient autonomy to be termed "selfish". Their robustness is linked to quality control mechanisms which are embedded in and act upon the individual networks, thereby providing a basis for selection during gametogenesis. These selective processes are equally likely to affect cellular functions that are not gamete-specific, and the evolution of the most complex organisms, including man, is therefore likely to occur via two pathways: essential housekeeping functions would be regulated and evolve during gametogenesis within the parents before being transmitted to their progeny, while classical selection would operate on other traits of the organisms that shape their fitness with respect to the environment.     

Mathematical descriptions of biochemical networks: stability, stochasticity, evolution

In this paper, we review some fundamental aspects, as well as some new developments, in the emerging field of network biology. The focus of attention is placed on mathematical approaches to conceptual modeling of biomolecular networks with special emphasis on dynamic stability, stochasticity and evolution.     

Quantitative analysis of signaling networks

The response of biological cells to environmental change is coordinated by protein-based signaling networks. These networks are to be found in both prokaryotes and eukaryotes. In eukaryotes, the signaling networks can be highly complex, some networks comprising of 60 or more proteins. The fundamental motif that has been found in all signaling networks is the protein phosphorylation/dephosphorylation cycle--the cascade cycle. At this time, the computational function of many of the signaling networks is poorly understood. However, it is clear that it is possible to construct a huge variety of control and computational circuits, both analog and digital from combinations of the cascade cycle. In this review, we will summarize the great versatility of the simple cascade cycle as a computational unit and towards the end give two examples, one prokaryotic chemotaxis circuit and the other, the eukaryotic MAPK cascade.     

Propensity of a circadian clock to adjust to the 24h day-night light cycle and its sensitivity to molecular noise

The circadian clock of Drosophila melanogaster and its tendency to adjust to the day-night light cycle is simulated by deterministic and stochastic methods. The robustness of the locking to the light-cycle with respect to molecular noise is studied. It is found that within the model studied, the molecular noise in the stochastic simulation erases the finer injection-locking structures, stronger injection signals are needed and the locking has the character of prolonged locked time intervals with cycle slips in between. The simulations are compared to a simple injection-locking model with noise that seems to describe the overall behavior well.     

Design principles for robust biochemical reaction networks: what works, what cannot work, and what might almost work

We bring together recent results that connect the structure of a mass-action reaction network to its capacity for concentration robustness — that is, its capacity to keep the concentration of a critical bio-active species within narrow limits, even against large fluctuations in the overall supply of the network’s constituents.     

Characterizing robustness and sensitivity of convolutional neural networks for quantitative analysis of mitochondrial morphology

Background: Quantitative analysis of mitochondrial morphology plays important roles in studies of mitochondrial biology. The analysis depends critically on segmentation of mitochondria, the image analysis process of extracting mitochondrial morphology from images. The main goal of this study is to characterize the performance of convolutional neural networks (CNNs) in segmentation of mitochondria from fluorescence microscopy images. Recently, CNNs have achieved remarkable success in challenging image segmentation tasks in several disciplines. So far, however, our knowledge of their performance in segmenting biological images remains limited. In particular, we know little about their robustness, which defines their capability of segmenting biological images of different conditions, and their sensitivity, which defines their capability of detecting subtle morphological changes of biological objects. Methods: We have developed a method that uses realistic synthetic images of different conditions to characterize the robustness and sensitivity of CNNs in segmentation of mitochondria. Using this method, we compared performance of two widely adopted CNNs: the fully convolutional network (FCN) and the U-Net. We further compared the two networks against the adaptive active-mask (AAM) algorithm, a representative of high-performance conventional segmentation algorithms. Results: The FCN and the U-Net consistently outperformed the AAM in accuracy, robustness, and sensitivity, often by a significant margin. The U-Net provided overall the best performance. Conclusions: Our study demonstrates superior performance of the U-Net and the FCN in segmentation of mitochondria. It also provides quantitative measurements of the robustness and sensitivity of these networks that are essential to their applications in quantitative analysis of mitochondrial morphology.     

History of protein-protein interactions: from egg-white to complex networks

Today, it is widely appreciated that protein-protein interactions play a fundamental role in biological processes. This was not always the case. The study of protein interactions started slowly and evolved considerably, together with conceptual and technological progress in different areas of research through the late 19th and the 20th centuries. In this review, we present some of the key experiments that have introduced major conceptual advances in biochemistry and molecular biology, and review technological breakthroughs that have paved the way for today's systems-wide approaches to protein-protein interaction analysis.     

Bayesian Calibration of a Stochastic Kinetic Computer Model Using Multiple Data Sources

Summary .  In this article, we describe a Bayesian approach to the calibration of a stochastic computer model of chemical kinetics. As with many applications in the biological sciences, the data available to calibrate the model come from different sources. Furthermore, these data appear to provide somewhat conflicting information about the model parameters. We describe a modeling framework that allows us to synthesize this conflicting information and arrive at a consensus inference. In particular, we show how random effects can be incorporated into the model to account for between-individual heterogeneity that may be the source of the apparent conflict.     

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms

We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24h sleep-wake syndrome.     

Control of Boolean networks: hardness results and algorithms for tree structured networks

Finding control strategies of cells is a challenging and important problem in the post-genomic era. This paper considers theoretical aspects of the control problem using the Boolean network (BN), which is a simplified model of genetic networks. It is shown that finding a control strategy leading to the desired global state is computationally intractable (NP-hard) in general. Furthermore, this hardness result is extended for BNs with considerably restricted network structures. These results justify existing exponential time algorithms for finding control strategies for probabilistic Boolean networks (PBNs). On the other hand, this paper shows that the control problem can be solved in polynomial time if the network has a tree structure. Then, this algorithm is extended for the case where the network has a few loops and the number of time steps is small. Though this paper focuses on theoretical aspects, biological implications of the theoretical results are also discussed.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

A Bayesian framework for the analysis of cospeciation

Abstract.— Information on the history of cospeciation and host switching for a group of host and parasite species is contained in the DNA sequences sampled from each. Here, we develop a Bayesian framework for the analysis of cospeciation. We suggest a simple model of host switching by a parasite on a host phylogeny in which host switching events are assumed to occur at a constant rate over the entire evolutionary history of associated hosts and parasites. The posterior probability density of the parameters of the model of host switching are evaluated numerically using Markov chain Monte Carlo. In particular, the method generates the probability density of the number of host switches and of the host switching rate. Moreover, the method provides information on the probability that an event of host switching is associated with a particular pair of branches. A Bayesian approach has several advantages over other methods for the analysis of cospeciation. In particular, it does not assume that the host or parasite phylogenies are known without error; many alternative phylogenies are sampled in proportion to their probability of being correct.     

Dynamical analysis of cellular networks based on the Green's function matrix

The complexity of cellular networks often limits human intuition in understanding functional regulations in a cell from static network diagrams. To this end, mathematical models of ordinary differential equations (ODEs) have commonly been used to simulate dynamical behavior of cellular networks, to which a quantitative model analysis can be applied in order to gain biological insights. In this paper, we introduce a dynamical analysis based on the use of Green's function matrix (GFM) as sensitivity coefficients with respect to initial concentrations. In contrast to the classical (parametric) sensitivity analysis, the GFM analysis gives a dynamical, molecule-by-molecule insight on how system behavior is accomplished and complementarily how (impulse) signal propagates through the network. The knowledge gained will have application from model reduction and validation to drug discovery research in identifying potential drug targets, studying drug efficacy and specificity, and optimizing drug dosing and timing. The efficacy of the method is demonstrated through applications to common network motifs and a Fas-induced programmed cell death model in Jurkat T cell line.     

Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to non-steady state trajectories

A sensitivity analysis of general stoichiometric networks is considered. The results are presented as a generalization of Metabolic Control Analysis, which has been concerned primarily with system sensitivities at steady state. An expression for time-varying sensitivity coefficients is given and the Summation and Connectivity Theorems are generalized. The results are compared to previous treatments. The analysis is accompanied by a discussion of the computation of the sensitivity coefficients and an application to a model of phototransduction.     

Extinction of marine renewable resources: a demographic analysis

A simple (deterministic) population dynamics model is used to examine the level of fishing mortality at which a population is expected to be driven to extinction (F _crash). Values for F _crash are determined for six marine fish species (Cape hake, blue grenadier, blue warehou, school shark, gummy shark, and orange roughy) subject to commercial harvest. The sensitivity of the value of F _crash to changing the selectivity pattern of the fishery and the relationship between the size of the reproductive component of the population and subsequent births to allow for depensatory effects is examined. F _crash is greatest for highly productive species and when the fishery does not target immature animals. The ratio of F _crash to the fishing mortality at which maximum sustainable yield is achieved, F _MSY, is a decreasing function of the productivity of the population. The possibility of depensation has little impact on F _MSY but can substantially reduce the ratio F _crash/F _MSY. A series of stochastic simulations is conducted to assess the probability of detecting that the extent of fishing exceeds F _crash using the current IUCN A criterion. The results of these simulations are also used to determine the probability that the IUCN A criteria will be triggered when fishing takes place at F _MSY. The results indicate that there are substantial probabilities of incorrectly identifying species being harvested at F _MSY as being threatened during the 'fishing down' phase and also of not identifying species actually at risk of extinction if fishing mortality is not reduced. Received: January 14, 1999 / Accepted: April 27, 1999     

The role of dynamic stimulation pattern in the analysis of bistable intracellular networks

Bistable systems play an important role in the functioning of living cells. Depending on the strength of the necessary positive feedback one can distinguish between (irreversible) “one-way switch” or (reversible) “toggle-switch” type behavior. Besides the well- established steady-state properties, some important characteristics of bistable systems arise from an analysis of their dynamics. We demonstrate that a supercritical stimulus amplitude is not sufficient to move the system from the lower (off-state) to the higher branch (on-state) for either a step or a pulse input. A switching surface is identified for the system as a function of the initial condition, input pulse amplitude and duration (a supercritical signal). We introduce the concept of bounded autonomy for single level systems with a pulse input. Towards this end, we investigate and characterize the role of the duration of the stimulus. Furthermore we show, that a minimal signal power is also necessary to change the steady state of the bistable system. This limiting signal power is independent of the applied stimulus and is determined only by systems parameters. These results are relevant for the design of experiments, where it is often difficult to create a defined pattern for the stimulus. Furthermore, intracellular processes, like receptor internalization, do manipulate the level of stimulus such that level and duration of the stimulus is conducive to characteristic behavior.     

Assessment of effects of experimental imprecision on optimized biochemical systems.

Metabolic pathways may be optimized with S-system models that prescribe profiles of control variables leading to optimal output while keeping metabolites and enzyme activities within predefined ranges. Monte Carlo simulations show how much the yield and the corresponding metabolite concentrations would be affected by inaccuracies in the experimental implementation of the prescribed profiles. For a recent model of citric acid production in Aspergillus niger, the yield is roughly normally distributed, whereas the distributions of metabolite concentrations differ greatly in shape and statistical characteristics. Even moderate inaccuracies may lead to constraint violations, which appear to be correlated with high logarithmic gains.