Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

Turing’s pattern formation mechanism exhibits sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Nonetheless, secondary aspects of developmental self-organisation, such as a growing domain, have been shown to ameliorate this aberrant model behaviour. Furthermore, while in-situ hybridisation reveals the presence of gene expression in developmental processes, the influence of such dynamics on Turing’s model has received limited attention. Here, we novelly focus on the Gierer–Meinhardt reaction diffusion system considering delays due the time taken for gene expression, while incorporating a number of different domain growth profiles to further explore the influence and interplay of domain growth and gene expression on Turing’s mechanism. We find extensive pathological model behaviour, exhibiting one or more of the following: temporal oscillations with no spatial structure, a failure of the Turing instability and an extreme sensitivity to the initial conditions, the growth profile and the duration of gene expression. This deviant behaviour is even more severe than observed in previous studies of Schnakenberg kinetics on exponentially growing domains in the presence of gene expression (Gaffney and Monk in Bull. Math. Biol. 68:99–130, 2006). Our results emphasise that gene expression dynamics induce unrealistic behaviour in Turing’s model for multiple choices of kinetics and thus such aberrant modelling predictions are likely to be generic. They also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing’s mechanism in the presence of gene expression time delays. The above, extensive, pathologies suggest that, in the presence of gene expression, Turing’s mechanism would generally require a novel and extensive secondary mechanism to control reaction diffusion patterning.     

A modular approach for modeling the cell cycle based on functional response curves

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.     

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.     

Stochastic Models of Gene Expression with Delayed Degradation

In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.We propose a new methodology to deal with time delays in biological systems and apply it to simple models of gene expression with delayed degradation. We show that time delay of protein degradation does not cause oscillations as it was recently argued. It follows from our rigorous analysis that one should look for different mechanisms responsible for oscillations observed in biological experiments.We develop a systematic analytical treatment of stochastic models of time delays. Specifically we take into account that some reactions, for example degradation, are consuming, that is: once molecules start to degrade they cannot be part in other degradation processes.We introduce an auxiliary stochastic process and calculate analytically the variance and the autocorrelation function of the number of protein molecules in stationary states in basic models of delayed protein degradation.     

Grasshopper hunchback expression reveals conserved and novel aspects of axis formation and segmentation

While the expression patterns of segment polarity genes such as engrailed have been shown to be similar in Drosophila melanogaster and Schistocerca americana (grasshopper), the expression patterns of pair-rule genes such as even-skipped are not conserved between these species. This might suggest that the factors upstream of pair-rule gene expression are not conserved across insect species. We find that, despite this, many aspects of the expression of the Drosophila gap gene hunchback are shared with its orthologs in the grasshoppers S. americana and L. migratoria. We have analyzed both mRNA and protein expression during development, and find that the grasshopper hunchback orthologs appear to have a conserved role in early axial patterning of the germ anlagen and in the specification of gnathal and thoracic primordia. In addition, distinct stepped expression levels of hunchback in the gnathal/thoracic domains suggest that grasshopper hunchback may act in a concentration-dependent fashion (as in Drosophila), although morphogenetic activity is not set up by diffusion to form a smooth gradient. Axial patterning functions appear to be performed entirely by zygotic hunchback, a fundamental difference from Drosophila in which maternal and zygotic hunchback play redundant roles. In grasshoppers, maternal hunchback activity is provided uniformly to the embryo as protein and, we suggest, serves a distinct role in distinguishing embryonic from extra-embryonic cells along the anteroposterior axis from the outset of development - a distinction made in Drosophila along the dorsoventral axis later in development. Later hunchback expression in the abdominal segments is conserved, as are patterns in the nervous system, and in both Drosophila and grasshopper, hunchback is expressed in a subset of extra-embryonic cells. Thus, while the expected domains of hunchback expression are conserved in Schistocerca, we have found surprising and fundamental differences in axial patterning, and have identified a previously unreported domain of expression in Drosophila that suggests conservation of a function in extra-embryonic patterning.     

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer–Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.     

Cyp26 enzymes generate the retinoic acid response pattern necessary for hindbrain development

Retinoic acid (RA) is essential for normal vertebrate development, including the patterning of the central nervous system. During early embryogenesis, RA is produced in the trunk mesoderm through the metabolism of vitamin A derived from the maternal diet and behaves as a morphogen in the developing hindbrain where it specifies nested domains of Hox gene expression. The loss of endogenous sources of RA can be rescued by treatment with a uniform concentration of exogenous RA, indicating that domains of RA responsiveness can be shaped by mechanisms other than the simple diffusion of RA from a localized posterior source. Here, we show that the cytochrome p450 enzymes of the Cyp26 class, which metabolize RA into polar derivatives, function redundantly to shape RA-dependent gene-expression domains during hindbrain development. In zebrafish embryos depleted of the orthologs of the three mammalian CYP26 genes CYP26A1, CYP26B1 and CYP26C1, the entire hindbrain expresses RA-responsive genes that are normally restricted to nested domains in the posterior hindbrain. Furthermore, we show that Cyp26 enzymes are essential for exogenous RA to rescue hindbrain patterning in RA-depleted embryos. We present a ;gradient-free' model for hindbrain patterning in which differential RA responsiveness along the hindbrain anterior-posterior axis is shaped primarily by the dynamic expression of RA-degrading enzymes.     

Numerical bifurcation analysis of delay differential equations arising from physiological modeling

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.     

Pre-steady-state decoding of the Bicoid morphogen gradient

Morphogen gradients are established by the localized production and subsequent diffusion of signaling molecules. It is generally assumed that cell fates are induced only after morphogen profiles have reached their steady state. Yet, patterning processes during early development occur rapidly, and tissue patterning may precede the convergence of the gradient to its steady state. Here we consider the implications of pre-steady-state decoding of the Bicoid morphogen gradient for patterning of the anterior–posterior axis of the Drosophila embryo. Quantitative analysis of the shift in the expression domains of several Bicoid targets (gap genes) upon alteration of bcd dosage, as well as a temporal analysis of a reporter for Bicoid activity, suggest that a transient decoding mechanism is employed in this setting. We show that decoding the pre-steady-state morphogen profile can reduce patterning errors caused by fluctuations in the rate of morphogen production. This can explain the surprisingly small shifts in gap and pair-rule gene expression domains observed in response to alterations in bcd dosage.     

Hedgehog signalling from the zona limitans intrathalamica orchestrates patterning of the zebrafish diencephalon

Midway between the anterior neural border and the midbrain-hindbrain boundary, two well-known local signalling centres in the early developing brain, is a further transverse boundary with putative signalling properties -- the zona limitans intrathalamica (ZLI). Here, we describe formation of the ZLI in zebrafish in relation to expression of sonic hedgehog (shh) and tiggy-winkle hedgehog (twhh), and to development of the forebrain regions that flank the ZLI: the prethalamus and thalamus. We find that enhanced Hh signalling increases the size of prethalamic and thalamic gene expression domains, whereas lack of Hh signalling leads to absence of these domains. In addition, we show that shh and twhh display both unique and redundant functions during diencephalic patterning. Genetic ablation of the basal plate shows that Hh expression in the ZLI alone is sufficient for diencephalic differentiation. Furthermore, acquisition of correct prethalamic and thalamic gene expression is dependent on direct Hh signalling. We conclude that proper maturation of the diencephalon requires ZLI-derived Hh signalling.     

Tbx5 and Tbx20 act synergistically to control vertebrate heart morphogenesis

Members of the T-box family of proteins play a fundamental role in patterning the developing vertebrate heart; however, the precise cellular requirements for any one family member and the mechanism by which individual T-box genes function remains largely unknown. In this study, we have investigated the cellular and molecular relationship between two T-box genes, Tbx5 and Tbx20. We demonstrate that blocking Tbx5 or Tbx20 produces phenotypes that display a high degree of similarity, as judged by overall gross morphology, molecular marker analysis and cardiac physiology, implying that the two genes are required for and have non-redundant functions in early heart development. In addition, we demonstrate that although co-expressed, Tbx5 and Tbx20 are not dependent on the expression of one another, but rather have a synergistic role during early heart development. Consistent with this proposal, we show that TBX5 and TBX20 can physically interact and map the interaction domains, and we show a cellular interaction for the two proteins in cardiac development, thus providing the first evidence for direct interaction between members of the T-box gene family.     

扩散与时滞有关的单种群植物模型的全局稳定性

研究了具有周期系数及扩散与时滞有关的单种群植物模型,得到了该系统存在全局渐近稳定的正周期解的充分条件,其中该条件依赖时滞与扩散