A model of morphogenetic pattern formation

A model for the morphogenetic movement of surfaces composed of cellular monolayers is proposed. The cells are presumed joined at their lateral surfaces. An otherwise unspecified substance called a "morphogen" is introduced which is the agent of change in the individual cell (or cell-like region). The distribution of these cellular deformations define a surface (the middle surface, through the middle of the cell heights) via equations given for the Gauss and Mean curvatures of the surface defined at each point. The Gauss curvature as a function of the morphogen level determines the metric of the surface "g(u, v)" in conformal co-ordinates u, v. A unique equation for the morphogen distribution over the survace is presented which has the property of size invariance, that is, the model "regulates" without need of further arguments. The two resulting coupled equations for the metric and the morphogen, eqns (4) and (2), both non-linear equations, are to be solved self-consistently, once the individual cell deformation as a function of morphogen is given. The surface geometry determines the morphogen distribution, and the morphogen distribution in turn affects the surface geometry. Extension of the model to two or more morphogens is straightforward, and the key property of "regulation" or size invariance of the model is retained. Numerical integration of the two coupled equations is carried out in the case of axial symmetry, and the results presented by the case that individual cells deform by changing the ratio of their apical to basal areas, as well as their heights. Gastrulation in small regulating holoblastic eggs (e.g. starfish, sea urchin and amphioxus) is discussed in light of the present model.     

A mathematical model of metabolic insulin signaling pathways

We develop a mathematical model that explicitly represents many of the known signaling components mediating translocation of the insulin-responsive glucose transporter GLUT4 to gain insight into the complexities of metabolic insulin signaling pathways. A novel mechanistic model of postreceptor events including phosphorylation of insulin receptor substrate-1, activation of phosphatidylinositol 3-kinase, and subsequent activation of downstream kinases Akt and protein kinase C-zeta is coupled with previously validated subsystem models of insulin receptor binding, receptor recycling, and GLUT4 translocation. A system of differential equations is defined by the structure of the model. Rate constants and model parameters are constrained by published experimental data. Model simulations of insulin dose-response experiments agree with published experimental data and also generate expected qualitative behaviors such as sequential signal amplification and increased sensitivity of downstream components. We examined the consequences of incorporating feedback pathways as well as representing pathological conditions, such as increased levels of protein tyrosine phosphatases, to illustrate the utility of our model for exploring molecular mechanisms. We conclude that mathematical modeling of signal transduction pathways is a useful approach for gaining insight into the complexities of metabolic insulin signaling.     

STDP in a bistable synapse model based on CaMKII and associated signaling pathways

The calcium/calmodulin-dependent protein kinase II (CaMKII) plays a key role in the induction of long-term postsynaptic modifications following calcium entry. Experiments suggest that these long-term synaptic changes are all-or-none switch-like events between discrete states. The biochemical network involving CaMKII and its regulating protein signaling cascade has been hypothesized to durably maintain the evoked synaptic state in the form of a bistable switch. However, it is still unclear whether experimental LTP/LTD protocols lead to corresponding transitions between the two states in realistic models of such a network. We present a detailed biochemical model of the CaMKII autophosphorylation and the protein signaling cascade governing the CaMKII dephosphorylation. As previously shown, two stable states of the CaMKII phosphorylation level exist at resting intracellular calcium concentration, and high calcium transients can switch the system from the weakly phosphorylated (DOWN) to the highly phosphorylated (UP) state of the CaMKII (similar to a LTP event). We show here that increased CaMKII dephosphorylation activity at intermediate Ca²⁺ concentrations can lead to switching from the UP to the DOWN state (similar to a LTD event). This can be achieved if protein phosphatase activity promoting CaMKII dephosphorylation activates at lower Ca²⁺ levels than kinase activity. Finally, it is shown that the CaMKII system can qualitatively reproduce results of plasticity outcomes in response to spike-timing dependent plasticity (STDP) and presynaptic stimulation protocols. This shows that the CaMKII protein network can account for both induction, through LTP/LTD-like transitions, and storage, due to its bistability, of synaptic changes.     

A cellular oscillator model for periodic pattern formation

In this paper, we present a model for pattern formation in developing organisms that is based on cellular oscillators (CO). An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell autonomous or non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. We apply a CO model to vertebrate somitogenesis and show that we can reproduce the dynamics of periodic gene expression patterns observed in the pre-somitic mesoderm. We also show how varying somite lengths can be generated with the CO model. We then discuss the model in view of experimental evidence and its relevance to other instances of biological pattern formation, showing its versatility as a pattern generator.     

A mouse model to identify cooperating signaling pathways in cancer

We here establish a mouse cancer model called Multi-Hit that allows for the evaluation of oncogene cooperativities in tumor development. The model is based on the stochastic expression of oncogene combinations ('hits') that are mediated by Cre in a given tissue. Cells with cooperating hits are positively selected and give rise to tumors. We used this approach to evaluate the requirement of Ras downstream effector pathways in tumorigenesis.     

Generalized logical model based on network topology to capture the dynamical trends of cellular signaling pathways

Background

Cellular responses to extracellular perturbations require signaling pathways to capture and transmit the signals. However, the underlying molecular mechanisms of signal transduction are not yet fully understood, thus detailed and comprehensive models may not be available for all the signaling pathways. In particular, insufficient knowledge of parameters, which is a long-standing hindrance for quantitative kinetic modeling necessitates the use of parameter-free methods for modeling and simulation to capture dynamic properties of signaling pathways.

Results

We present a computational model that is able to simulate the graded responses to degradations, the sigmoidal biological relationships between signaling molecules and the effects of scheduled perturbations to the cells. The simulation results are validated using experimental data of protein phosphorylation, demonstrating that the proposed model is capable of capturing the main trend of protein activities during the process of signal transduction. Compared with existing simulators, our model has better performance on predicting the state transitions of signaling networks.

Conclusion

The proposed simulation tool provides a valuable resource for modeling cellular signaling pathways using a knowledge-based method.

     

Structural modeling and analysis of signaling pathways based on Petri nets

The purpose of this paper is to discuss how to model and analyze signaling pathways by using Petri net. Firstly, we propose a modeling method based on Petri net by paying attention to the molecular interactions and mechanisms. Then, we introduce a new notion "activation transduction component" in order to describe an enzymic activation process of reactions in signaling pathways and shows its correspondence to a so-called elementary T-invariant in the Petri net models. Further, we design an algorithm to effectively find basic enzymic activation processes by obtaining a series of elementary T-invariants in the Petri net models. Finally, we demonstrate how our method is practically used in modeling and analyzing signaling pathway mediated by thrombopoietin as an example.     

Integrin-specific signaling pathways controlling focal adhesion formation and cell migration

The fibronectin (FN)-binding integrins alpha4beta1 and alpha5beta1 confer different cell adhesive properties, particularly with respect to focal adhesion formation and migration. After analyses of alpha4+/alpha5+ A375-SM melanoma cell adhesion to fragments of FN that interact selectively with alpha4beta1 and alpha5beta1, we now report two differences in the signals transduced by each receptor that underpin their specific adhesive properties. First, alpha5beta1 and alpha4beta1 have a differential requirement for cell surface proteoglycan engagement for focal adhesion formation and migration; alpha5beta1 requires a proteoglycan coreceptor (syndecan-4), and alpha4beta1 does not. Second, adhesion via alpha5beta1 caused an eightfold increase in protein kinase Calpha (PKCalpha) activation, but only basal PKCalpha activity was observed after adhesion via alpha4beta1. Pharmacological inhibition of PKCalpha and transient expression of dominant-negative PKCalpha, but not dominant-negative PKCdelta or PKCzeta constructs, suppressed focal adhesion formation and cell migration mediated by alpha5beta1, but had no effect on alpha4beta1. These findings demonstrate that different integrins can signal to induce focal adhesion formation and migration by different mechanisms, and they identify PKCalpha signaling as central to the functional differences between alpha4beta1 and alpha5beta1.     

Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.     

rugose (rg), a Drosophila A kinase anchor protein,is required for retinal pattern formation and interacts genetically with multiple signaling pathways

In the developing Drosophila eye, cell fate determination and pattern formation are directed by cell-cell interactions mediated by signal transduction cascades. Mutations at the rugose locus (rg) result in a rough eye phenotype due to a disorganized retina and aberrant cone cell differentiation, which leads to reduction or complete loss of cone cells. The cone cell phenotype is sensitive to the level of rugose gene function. Molecular analyses show that rugose encodes a Drosophila A kinase anchor protein (DAKAP 550). Genetic interaction studies show that rugose interacts with the components of the EGFR- and Notch-mediated signaling pathways. Our results suggest that rg is required for correct retinal pattern formation and may function in cell fate determination through its interactions with the EGFR and Notch signaling pathways.     

A mathematical model of pattern formation in the vascular cambium of trees

The beautiful patterns apparent in wood grain have their origin in the alignment of fusiform initial cells in the vascular cambium of trees. We develop a mathematical model to describe the orientation of fusiform initial cells, and their interaction with the plant hormone indole-3-acetic acid (auxin). The model incorporates the following four assumptions: (1) auxin is actively transported parallel to the long axis of the initials, (2) auxin diffuses perpendicular to the long axis of the initials, (3) the initials tend to orient parallel to the flux of auxin through the cambium, and (4) adjacent initials tend to orient parallel to one another. Each assumption is justified on the basis of available evidence and cast in mathematical form. Our main result is a pair of nonlinear differential equations that describe the coupling between the distribution of auxin in the cambium and the orientation of fusiform initials. Numerical solutions to the equations show qualitative resemblance to the wood grain patterns observed at branch junctions, wounds and knots, and topological defects.     

A cell pattern generation model based on an extended artificial regulatory network

Cell pattern generation has a fundamental role in both artificial and natural development. This paper presents results from a model in which a genetic algorithm (GA) was used to evolve an artificial regulatory network (ARN) to produce predefined 2D cell patterns through the selective activation and inhibition of genes. The ARN used in this work is an extension of a model previously used to create simple geometrical patterns. The GA worked by evolving the gene regulatory network that was used to control cell reproduction, which took place in a testbed based on cellular automata (CA). After the final chromosomes were produced, a single cell in the middle of the CA lattice was allowed to replicate controlled by the ARN found by the GA, until the desired cell pattern was formed. The model was applied to the problem of generating a French flag pattern.     

Spermidine regulates Vibrio cholerae biofilm formation via transport and signaling pathways

Vibrio cholerae , the causative agent of the devastating diarrheal disease cholera, can form biofilms on diverse biotic and abiotic surfaces. Biofilm formation is important for the survival of this organism both in its natural environment and in the human host. Development of V. cholerae biofilms are regulated by complex regulatory networks that respond to environmental signals. One of these signals, norspermidine, is a polyamine that enhances biofilm formation via the NspS/MbaA signaling system. In this work, we have investigated the role of the polyamine spermidine in regulating biofilm formation in V. cholerae . We show that spermidine import requires PotD1, an ortholog of the periplasmic substrate-binding protein of the spermidine transport system in Escherichia coli . We also show that deletion of the potD1 gene results in a significant increase in biofilm formation. We hypothesize that spermidine imported into the cell hinders biofilm formation. Exogenous spermidine further reduces biofilm formation in a PotD1-independent, but NspS/MbaA-dependent, manner. Our results suggest that polyamines affect biofilm formation in V. cholerae via multiple pathways involving both transport and signaling networks.     

Ontology based standardization of Petri net modeling for signaling pathways

Taking account of the great availability of Petri nets in modeling and analyzing large complicated signaling networks, semantics of Petri nets is in need of systematization for the purpose of consistency and reusability of the models. This paper reports on standardization of units of Petri nets on the basis of an ontology that gives an intrinsic definition to the process of signaling in signaling pathways.     

A model of pattern formation in the primary skin follicle population of sheep

Wool follicles in the skin of sheep are arranged in groups. Characteristic two-dimensional spatial relationships exist between the groups as well as between the follicles within each group. The developmental origin of primary follicle patterns is related to a simple physiological model employing diffusion as one of its major components. A computer study of the deterministic properties of the model shows it to give a satisfactory explanation of the pattern of primary follicles.