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-locu

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

<p>Abstract</p> <p>Background</p> <p>Genetic variation explains a considerable part of observed phenotypic variation in gene expression networks. This variation has been shown to be located both locally (<it>cis</it>) and distally (<it>trans</it>) 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.</p> <p>Results</p> <p>By 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 <it>cis</it>/<it>trans </it>linkage patterns. However, when the shape of the <it>cis-</it>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.</p> <p>Conclusion</p> <p>Our findings indicate that genetic variation affecting the form of <it>cis</it>-regulatory input functions may reshape the genotype-phenotype map by changing the relative importance of <it>cis </it>and <it>trans </it>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.</p

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

Background: A 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. Results: By 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. Conclusion: The 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

Threshold-dominated regulation hides genetic variation in gene expression networks

<p>Abstract</p> <p>Background</p> <p>In 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 (<it>CV</it>) of the gene product concentrations in the stable state for the regulating genes in absolute terms and compared to the <it>CV </it>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.</p> <p>Results</p> <p>For all the models, threshold robustness increases with increasing response steepness. The <it>CV</it>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.</p> <p>Conclusion</p> <p>Threshold 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.</p

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

Matematisk veiviser for fotgjengere

Fra Institutt for matematiske fag

Propagation of genetic variation in gene regulatory networks

A future quantitative genetics theory should link genetic variation to phenotypic variation in a causally cohesive way based on how genes actually work and interact. We provide a theoretical framework for predicting and understanding the manifestation of genetic variation in haploid and diploid regulatory networks with arbitrary feedback structures and intra-locus and inter-locus functional dependencies. Using results from network and graph theory, we define propagation functions describing how genetic variation in a locus is propagated through the network, and show how their derivatives are related to the network’s feedback structure. Similarly, feedback functions describe the effect of genotypic variation of a locus on itself, either directly or mediated by the network. A simple sign rule relates the sign of the derivative of the feedback function of any locus to the feedback loops involving that particular locus. We show that the sign of the phenotypically manifested interaction between alleles at a diploid locus is equal to the sign of the dominant feedback loop involving that particular locus, in accordance with recent results for a single locus system. Our results provide tools by which one can use observable equilibrium concentrations of gene products to disclose structural properties of the network architecture. Our work is a step towards a theory capable of explaining the pleiotropy and epistasis features of genetic variation in complex regulatory networks as functions of regulatory anatomy and functional location of the genetic variation

Elsevier Editorial System(tm) for Title: Dynamics of actively regulated gene networks Title: Dynamics of actively regulated gene networks Physica D: Nonlinear Phenomena Dynamics of actively regulated gene networks

we have just re-submitted the paper above on the Elsevier journal web site. We have revised it in accordance with the reviewers&apos; comments. In the letter to reviewers (a file uploaded to the submission web site), each point raised has been discussed, and the consequent possible changes in the manuscript have been pointed out. Please, note that in the revised version new references have been added. We are now quite confident that the paper should be ready for publication as all the objections of the referees have been met in the new version (with a few exception in which we mean the referee was wrong). Abstract A popular way of modelling gene regulatory networks is to design the models as if the genes were acting directly on each other. Genes are activated or inhibited by transcription factors which are direct gene products. The action of a transcription factor on a gene is modelled as a binary on-off dose-response function or a sigmoid around a certain threshold concentration. A transcription factor may act on several genes with equal or different thresholds. The combined effect of several transcription factors on a gene is generally assumed to obey Boolean-like composition rules. A mathematical challenge related to such network models is to analyse the behaviour in the narrow domains in phase space where at least one variable is close to one of its thresholds, called switching domains. In the present paper we analyse the motion in switching domains for models with steep sigmoidal response functions under the assumption that each transcription factor only regulates one gene at each of its thresholds. This biologically reasonable assumption allows us to establish rules that determine the passage through any switching domain and the sequence of domains through which the system passes. The rules can be effectively implemented in a sound automated analyser for such networks. The assumption simplifies certain mathematical and computational issues, but unfortunately may be at variance with one of the necessary conditions for the singular perturbation theory which our derivations are based on. We give sufficient conditions ensuring that, despite this, the difficulties raised by our assumption can be evaded and singular perturbation approximation safely applied. Keywords: Gene regulatory network, active regulation, sigmoidal response, singular perturbation. 82.39.Rt, 87.10.Ed, 87.18.Cf, 87.18 We divide our response into three parts: PACS I think it is important to provide more justification for the claim that this class actually captures a reasonable number of real biological networks (if not, do we really need to know this level of detail?). Transcription regulation is a very complicated process once you start studying it in molecular detail. *Revision Notes Our ref There is no general agreement on how to model transcription regulation in the most correct way. People choose different approaches: Boolean discrete models, piecewise linear ODE models, ODE models based on mass action law dynamics and/or Michaelis-Menten dynamics, Hill function with n = 2, some other value, either rather arbitrary or estimated from data, steep sigmoids, or combinations of these. There exist purely deterministic models as well as stochastic models. We are proposing models with steep sigmoids. In real systems some response functions could be steep, of course not all. Contrary to many other modelling schemes, ours can be analysed analytically, which is a great advantage. Combined models with steep sigmoids and other nonlinearities could certainly be made more realistic. Presently they can only be analysed in full numerically, although numerical investigations alone unsupported by analytical analyses are usually an unsatisfactory method, in particular for large and complex models, and not always possible without guesswork and unwarranted assumptions. See point 3 below. Steep sigmoid models may be useful in some cases, and our analysis may prove helpful as a starting point for analysing more complex models. This attitude implies that we do not claim that &quot;this class actually captures a reasonable number of real biological networks&quot;. But &quot;we really need to know this level of detail&quot; to lay a foundation for later generalisations and also to be able to subject the model framework to experimental tests. Unfortunately, we are not able to give modelling examples based on steep sigmoidal response functions apart from analyses of very small and simple networks. However, there are many examples of ODE models with step functions (piecewise linear models) and purely Boolean models, which both would appear to give more crude representations of the biological system. Thus, we believe the reason for the lack of larger sigmoidal models may be related to the fact that such models are more complicated to analyse than piecewise models. Also, in case the threshold of a TF for two different binding sites were equal, problems would arise in a purely deterministic model, as expounded by Cai et al. (our ref [23]). Different response profiles would make a concerted regulation of several genes at a common threshold virtually impossible. On the other hand, Cai et al. also showed that this problem can be resolved when one takes into account that transcription occurs in bursts. Then concerted regulation of many genes at the same threshold is possible, despite differences in response profiles. Thus, from this it seems that if we need to model 2 systems with switch-like regulation of several genes at a common threshold for a transcription factor, we would have to abandon determinism and include stochastisity or time-dependent variation in the transcription regulation. In addition coincident thresholds in steep sigmoid models generate mathematical difficulties. In Plahte and Kjøglum (our ref From this we conclude that in deterministic steep sigmoid models, no equal thresholds more likely provide useful models than common thresholds for several genes. For (1), there are indeed a few instances where gene-gene interactions can be well-represented by steep sigmoids (switch-like behaviour), but many more where the regulation is much smoother (graded). What happens in such instances? Can the analysis be extended to include a combination of graded and switch-like behaviours (e.g. a Goodwin oscillator model)? I don&apos;t believe there are too many instances where it is fair to represent everything as switch-like. We showed in Plahte and Kjøglum and in Veflingstad and Plahte (our refs [12, To incorporate these considerations in the paper we found it necessary to reorganise and partially rewrite the Introduction and add a few references (the ones mentioned above) and a few new paragraphs. Also, we have rearranged and partially rewritten the beginning of Sect. 5.6 to give a better explanation of the problem with applying singular perturbation under Assumption A.The last four sentences of the second paragraph in Sect. 7 has been deleted to avoid repetition. Response to Referee 2 We do thank this reviewer for the careful reading of the manuscript, and for his/her suggestions that helped us to make clearer the not immediate derivation of some points in the manuscript. The English/typing errors highlighted at points Point 6. We do not agree. The &quot;genes&quot; maintain a homeostatic state&quot;, not &quot;a set&quot;. Point 8. We have asked an English mother language person who goes for &quot;an&quot;. A web search didn&apos;t produce many hits for similar cases, but the one we found, also used &quot;an&quot;. So, we go for &quot;an&quot;. 3 Point 9. A switching variable is, per definition, a variable that is close to one of its thresholds. A variable x s cannot be close to more that one threshold because thresholds are not close to each other. However, this comment induced us to improve the point at issue. Point 18. Actually, the Lemma statement was a little ambiguous. We have changed it and now it should be clear that assumption A does not imply that J 0 SS is 1-2 block structured. This should also give an answer to the point 29. As for the last comment at this point, in accordance with the jargon in matrix perturbation analysis, it is standard, and then technically correct, to say that a matrix, or an eigenvalue, is stable. However, we added a sentence at page 8 in Section 4.1, para. 2 that specifies what we mean here by a stable matrix. Point 19. We have explained in the manuscript why multiple eigenvalues are not important for our analysis. Point 20. We have made the correction as requested, although, as we have pointed out at point 18, it can also be said &quot;stable eigenvalue&quot; (see point 18). Point 21. Yes, you are right. We have changed the sentence accordingly. Point 22 and Point 23. We have paraphrased the sentence and added information about the epsilons. Point 24. Actually, it was not clearly written, and we changed it as suggested. Now, it should be ok. Point 25. We have better detailed reference Point 33. We put this equation in two lines in accordance with the Instructions for Authors that require to break the equations, also in the first submission, so that the print in two columns is readable. Point 34. We have already said that we consider c(q) = 0 (see also point 19). At any rate, for c(q) = 0 the analysis is trivial (see, Abstract A popular way of modelling gene regulatory networks is to design the models as if the genes were acting directly on each other. Genes are activated or inhibited by transcription factors which are direct gene products. The action of a transcription factor on a gene is modelled as a binary on-off dose-response function or a sigmoid around a certain threshold concentration. A transcription factor may act on several genes with equal or different thresholds. The combined effect of several transcription factors on a gene is generally assumed to obey Boolean-like composition rules. A mathematical challenge related to such network models is to analyse the behaviour in the narrow domains in phase space where at least one variable is close to one of its thresholds, called switching domains. In the present paper we analyse the motion in switching domains for models with steep sigmoidal response functions under the assumption that each transcription factor only regulates one gene at each of its thresholds. This biologically reasonable assumption allows us to establish rules that determine the passage through any switching domain and the sequence of domains through which the system passes. The rules can be effectively implemented in a sound automated analyser for such networks. The assumption simplifies certain mathematical and computational issues, but unfortunately may be at variance with one of the necessary conditions for the singular perturbation theory which our derivations are based on. We give sufficient conditions ensuring that, despite this, the difficulties raised by our assumption can be evaded and singular perturbation approximation safely applied