Erik Plahte

FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS

By fairly simple considerations of stability and multistationarity in nonlinear systems of first order differential equations it is shown that under quite mild restrictions a negative feedback loop is a necessary condition for stability, and that a positive feedback loop is a necessary condition for multistationarity.

Astrocytic Mechanisms Explaining Neural-Activity-Induced Shrinkage of Extraneuronal Space

Neuronal stimulation causes ∼30% shrinkage of the extracellular space (ECS) between neurons and surrounding astrocytes in grey and white matter under experimental conditions. Despite its possible implications for a proper understanding of basic aspects of potassium clearance and astrocyte function, the phenomenon remains unexplained. Here we present a dynamic model that accounts for current experimental data related to the shrinkage phenomenon in wild-type as well as in gene knockout individuals. We find that neuronal release of potassium and uptake of sodium during stimulation, astrocyte uptake of potassium, sodium, and chloride in passive channels, action of the Na/K/ATPase pump, and osmotically driven transport of water through the astrocyte membrane together seem sufficient for generating ECS shrinkage as such. However, when taking into account ECS and astrocyte ion concentrations observed in connection with neuronal stimulation, the actions of the Na+/K+/Cl− (NKCC1) and the Na+/HCO3 − (NBC) cotransporters appear to be critical determinants for achieving observed quantitative levels of ECS shrinkage. Considering the current state of knowledge, the model framework appears sufficiently detailed and constrained to guide future key experiments and pave the way for more comprehensive astroglia–neuron interaction models for normal as well as pathophysiological situations.

Periodic solutions in systems of piecewise- linear differential equations

Motivated by the periodic behaviour of regulatory networks within cell biology and neurology, we have studied the periodic solutions of piecewise-linear, first- order differential equations with identical relative decay rates. The flow of the solution trajectories can be represented qualitatively by a directed graph. By examining the cycles in this graph and solving the eigenvalue problem for corresponding mapping matrices, all closed, period-1 orbits can be found by analytical means. Theorems about their exist- ence, stabiiiiy and uniqueness are derived. For three-dimensional systems, the basins of attraction of the limit cycles can be explicitly determined and it is shown that higher periodic and chaotic solutions do not exist.

Global analysis of steady points for systems of differential equations with sigmoid interactions

A new method to investigate asymptotic properties of linear differential equations with strong threshold and switching effects is presented. The method is applied to systems of equations of the form dx/dt = F(x) - yx, where y = constant and the dependence of F on x is mediated by sigmoid functions. Using a special sigmoid function called a logoid, which rises monotonically from zero to one in a narrow interval surrounding the threshold value, exact analytical expressions for the limiting value of all steady points can be found in the limit when the logoid approaches a step function. The limiting values are independent of the shape of the logoid for a large class of logoids. Relations between steady points and limit cycles of the equations with logoids, their step function limit and the corresponding piecewise linear equations are derived. It is found that the approximation of sigmoids by the step function idealization is not always warranted. The results strongly suggest the use of logoids instead of othe...

Threshold-dominated regulation hides genetic variation in gene expression networks

BackgroundIn dynamical models with feedback and sigmoidal response functions, some or all variables have thresholds around which they regulate themselves or other variables. A mathematical analysis has shown that when the dose-response functions approach binary or on/off responses, any variable with an equilibrium value close to one of its thresholds is very robust to parameter perturbations of a homeostatic state. We denote this threshold robustness. To check the empirical relevance of this phenomenon with response function steepnesses ranging from a near on/off response down to Michaelis-Menten conditions, we have performed a simulation study to investigate the degree of threshold robustness in models for a three-gene system with one downstream gene, using several logical input gates, but excluding models with positive feedback to avoid multistationarity. Varying parameter values representing functional genetic variation, we have analysed the coefficient of variation (CV) of the gene product concentrations in the stable state for the regulating genes in absolute terms and compared to the CV for the unregulating downstream gene. The sigmoidal or binary dose-response functions in these models can be considered as phenomenological models of the aggregated effects on protein or mRNA expression rates of all cellular reactions involved in gene expression.ResultsFor all the models, threshold robustness increases with increasing response steepness. The CV s of the regulating genes are significantly smaller than for the unregulating gene, in particular for steep responses. The effect becomes less prominent as steepnesses approach Michaelis-Menten conditions. If the parameter perturbation shifts the equilibrium value too far away from threshold, the gene product is no longer an effective regulator and robustness is lost. Threshold robustness arises when a variable is an active regulator around its threshold, and this function is maintained by the feedback loop that the regulator necessarily takes part in and also is regulated by. In the present study all feedback loops are negative, and our results suggest that threshold robustness is maintained by negative feedback which necessarily exists in the homeostatic state.ConclusionThreshold robustness of a variable can be seen as its ability to maintain an active regulation around its threshold in a homeostatic state despite external perturbations. The feedback loop that the system necessarily possesses in this state, ensures that the robust variable is itself regulated and kept close to its threshold. Our results suggest that threshold regulation is a generic phenomenon in feedback-regulated networks with sigmoidal response functions, at least when there is no positive feedback. Threshold robustness in gene regulatory networks illustrates that hidden genetic variation can be explained by systemic properties of the genotype-phenotype map.

The genotype-phenotype relationship in multicellular pattern-generating models - the neglected role of pattern descriptors

BackgroundA deep understanding of what causes the phenotypic variation arising from biological patterning processes, cannot be claimed before we are able to recreate this variation by mathematical models capable of generating genotype-phenotype maps in a causally cohesive way. However, the concept of pattern in a multicellular context implies that what matters is not the state of every single cell, but certain emergent qualities of the total cell aggregate. Thus, in order to set up a genotype-phenotype map in such a spatiotemporal pattern setting one is actually forced to establish new pattern descriptors and derive their relations to parameters of the original model. A pattern descriptor is a variable that describes and quantifies a certain qualitative feature of the pattern, for example the degree to which certain macroscopic structures are present. There is today no general procedure for how to relate a set of patterns and their characteristic features to the functional relationships, parameter values and initial values of an original pattern-generating model. Here we present a new, generic approach for explorative analysis of complex patterning models which focuses on the essential pattern features and their relations to the model parameters. The approach is illustrated on an existing model for Delta-Notch lateral inhibition over a two-dimensional lattice.ResultsBy combining computer simulations according to a succession of statistical experimental designs, computer graphics, automatic image analysis, human sensory descriptive analysis and multivariate data modelling, we derive a pattern descriptor model of those macroscopic, emergent aspects of the patterns that we consider of interest. The pattern descriptor model relates the values of the new, dedicated pattern descriptors to the parameter values of the original model, for example by predicting the parameter values leading to particular patterns, and provides insights that would have been hard to obtain by traditional methods.ConclusionThe results suggest that our approach may qualify as a general procedure for how to discover and relate relevant features and characteristics of emergent patterns to the functional relationships, parameter values and initial values of an underlying pattern-generating mathematical model.

Allele Interaction – Single Locus Genetics Meets Regulatory Biology

Background Since the dawn of genetics, additive and dominant gene action in diploids have been defined by comparison of heterozygote and homozygote phenotypes. However, these definitions provide little insight into the underlying intralocus allelic functional dependency and thus cannot serve directly as a mediator between genetics theory and regulatory biology, a link that is sorely needed. Methodology/Principal Findings We provide such a link by distinguishing between positive, negative and zero allele interaction at the genotype level. First, these distinctions disclose that a biallelic locus can display 18 qualitatively different allele interaction sign motifs (triplets of +, – and 0). Second, we show that for a single locus, Mendelian dominance is not related to heterozygote allele interaction alone, but is actually a function of the degrees of allele interaction in all the three genotypes. Third, we demonstrate how the allele interaction in each genotype is directly quantifiable in gene regulatory models, and that there is a unique, one-to-one correspondence between the sign of autoregulatory feedback loops and the sign of the allele interactions. Conclusion/Significance The concept of allele interaction refines single locus genetics substantially, and it provides a direct link between classical models of gene action and gene regulatory biology. Together with available empirical data, our results indicate that allele interaction can be exploited experimentally to identify and explain intricate intra- and inter-locus feedback relationships in eukaryotes.

STATIONARY STATES IN FOOD WEB MODELS WITH THRESHOLD RELATIONSHIPS

A new method for analysing stationary states in complex differential equation systems when the interaction terms contain sigmoid functions is presented. Originally aimed at simplifying the analysis of certain gene regulatory networks, the method is applicable to models comprising a wide range of sigmoid functions. The basic idea is to analyse the limiting case when the sigmoids approach the step function, and consider sigmoids with finite steepness as a perturbation. After a brief presentation the method is applied to a model for a herbivore feeding on two competing autotrophs. Analytical expresssions for the stationary points in the step function limit are given, and their dependence on the parameter values is analysed and interpreted.

Nonlinear regulation enhances the phenotypic expression of trans-acting genetic polymorphisms

BackgroundGenetic variation explains a considerable part of observed phenotypic variation in gene expression networks. This variation has been shown to be located both locally (cis) and distally (trans) to the genes being measured. Here we explore to which degree the phenotypic manifestation of local and distant polymorphisms is a dynamic feature of regulatory design.ResultsBy combining mathematical models of gene expression networks with genetic maps and linkage analysis we find that very different network structures and regulatory motifs give similar cis/trans linkage patterns. However, when the shape of the cis- regulatory input functions is more nonlinear or threshold-like, we observe for all networks a dramatic increase in the phenotypic expression of distant compared to local polymorphisms under otherwise equal conditions.ConclusionOur findings indicate that genetic variation affecting the form of cis-regulatory input functions may reshape the genotype-phenotype map by changing the relative importance of cis and trans variation. Our approach combining nonlinear dynamic models with statistical genetics opens up for a systematic investigation of how functional genetic variation is translated into phenotypic variation under various systemic conditions.

Monotonicity is a key feature of genotype-phenotype maps

It was recently shown that monotone gene action, i.e., order-preservation between allele content and corresponding genotypic values in the mapping from genotypes to phenotypes, is a prerequisite for achieving a predictable parent-offspring relationship across the whole allele frequency spectrum. Here we test the consequential prediction that the design principles underlying gene regulatory networks are likely to generate highly monotone genotype-phenotype maps. To this end we present two measures of the monotonicity of a genotype-phenotype map, one based on allele substitution effects, and the other based on isotonic regression. We apply these measures to genotype-phenotype maps emerging from simulations of 1881 different 3-gene regulatory networks. We confirm that in general, genotype-phenotype maps are indeed highly monotonic across network types. However, regulatory motifs involving incoherent feedforward or positive feedback, as well as pleiotropy in the mapping between genotypes and gene regulatory parameters, are clearly predisposed for generating non-monotonicity. We present analytical results confirming these deep connections between molecular regulatory architecture and monotonicity properties of the genotype-phenotype map. These connections seem to be beyond reach by the classical distinction between additive and non-additive gene action.