Tumor growth has been modeled at the macroscopic level by using established physical laws coupled with biological processes which are described in a phenomenological fashion. Such models consists of a system of PDEs in the tumor region; this region us unknown in advance, and thus its boundary is a "free boundary." In this talk I shall concentrate on the following questions: What are the shapes of dormant tumors? Are these shapes stable? The answer to these questions involve Liapounov - Schmidt and Hopf bifurcations for free boundary problems.
Much of natural resource management can be viewed as problems in spatial control: what to do, where to do it, when to do it, and how to monitor and assess the success of the effort. Despite the importance of this set of problems for numerous very practical matters (control of invasive species, epidemics of wildlife disease, water management, and setting hunting regulations), these problems are mostly addressed in a non-spatial manner, with generally no application of control in a spatial context. A variety of mathematical and computational approaches are available to address these problems, which I will summarize. I will then proceed to describe some of the more mathematical work ongoing with numerous collaborators (including Suzanne Lenhart, Scott Duke-Sylvester, Holly Gaff, Rene' Salinas and Andrew Whittle) to develop general theory for control of ecological systems in a spatial context. The ready availability of computatational capability opens up a variety of new opportunities for spatially-explicit control methods to be made accessible for resource managers concerned with site-specific issues as well as with regional-level coordination of effort. I argue that these problems present new opportunities for mathematicians to collaborate with computational scientists, natural resource managers, and geographers to develop a science of spatial control of natural systems.
A fundamental problem of cell biology is to understand how cells make measurements and then make behavioral decisions in response to these measurements. The full answer to this question is not known but there are some underlying principles that are coming to light. The short answer is that the rate of molecular diffusion contains quantifiable information that can be transduced by biochemical feedback to give control over physical structures.
This principle will be illustrated by two specific examples of how rates of molecular diffusion contain information that is used to make a measurement and a behavioral decision.
Example 1: Bacterial populations of P. aeruginosa are known to make a decision to secrete polymer gel and become virulent on the basis of the size of the colony in whch they live. This process is called quorum sensing and only recently has the mechanism for this been sorted out. It is now known that P. aeruginosa produces a chemical whose rate of diffusion out of the cell provides information about the size of the colony which when coupled with positive feedback gives rise to a hysteretic biochemical switch.
Example 2: Salmonella employ a mechanism that combines molecular diffusion with a negative feedback chemical network to "know" how long its flagella are. As a result, if a flagellum is cut off, it will regrow at the same rate with which it grew initially.
Mitosis is the process of segregation of chromosomes before cell division. This phenomenon is based on self-assembling molecular machine called mitotic spindle. The spindle consists of two asters made of microtubules, dynamic polymers that grow and shrink rapidly and repeatedly probing space and searching for the chromosomes. In the first part of the talk, I will address the problem of the self-organization of the aster considering experiments and models of fragments of fish melanophore cells that aggregate pigment granules coated with dynein molecular motors at the center. The motors move on microtubules, nucleate them and alter their dynamics. Elegant feedbacks cause beautiful pattern formation that helps to understand the asters formation. In the second part of the talk, I'll describe experiments and models of the 'Search and Capture' process and demonstrate that cells use chemical gradients to bias and optimize microtubule dynamics for fast division. I will demonstrate how a combination of analytical methods, computer simulations and quantitative experiments sheds light on principles of self-organization in cell division.
Mutualists and pathogens, collectively called symbionts, are ubiquitous in plant communities. While some symbionts are highly host-specific, others associate with multiple hosts. The outcome of multispecies host-symbionts interactions with different degrees of specificity are difficult to predict at this point due to a lack of a general conceptual framework. Complicating our predictive power is the fact that plant populations are spatially explicit and we know from past research that explicit space can profoundly alter plant-plant interactions. We introduce a spatially explicit, stochastic model to investigate the role of explicit space and host-specificity in multispecies host-symbiont interactions. We find that in our model, pathogens can significantly alter the spatial structure of plant communities, promoting coexistence, whereas mutualists appear to have only a limited effect. Effects are more pronounced the more host-specific symbionts are.
This talk describes joint work in progress with Boyce Griffith, David McQueen, and Edward Vigmond towards the construction of a combined electrical, mechanical, and fluid-mechanical model of the heart. The fiber architecture of the heart plays a central role in this model, and the mathematical principles governing cardiac fiber architecture will be discussed. It will then be shown how the immersed boundary method can be used in conjunction with a description of the fiber architecture to study both fluid-structure interaction and electrophysiology of the heart.
Strategies for new drug development have changed dramatically over recent years as the role of the functional dysregulation of protein interactions as the underlying cause of disease is increasingly understood. In the context of cancer, an expanding body of experimental and clinical literature attests to the promise of combination therapies which combine conventional treatments such as radiotherapy and chemotherapy with small molecule inhibitors that target activated kinase activity and protein-protein interactions in specific dysregulated pathways.
Nevertheless, a combination therapy that targets multiple interconnected nodes of a cell-signaling network represents a largely unexplored avenue in cancer treatment. Simple mathematical models of this novel 'network-targeted' combination therapy are presented here, pursuing the hypothesis that such an approach may induce the desired biochemical signal attenuation with lower doses of the necessary agents than when one node is targeted in isolation. The model demonstrates that attenuation of biochemical signals is significantly enhanced when multiple upstream processes are inhibited, in comparison with the inhibition of a single upstream process. Moreover, this enhanced attenuation is most pronounced in signals downstream of serially-connected target nodes. Importantly, the inhibition of serially-connected processes appears to have a synergistic (supra-additive) effect on the attenuation of downstream signals, owing to the highly non-linear relationships between ne! twork parameters and signals. These mathematical models also suggest that the topology of a cell-signaling network - its functional structure and the architecture of its control modules - has important implications for the most judicious choice of target nodes in a program of network-targeted combination therapy.
Since these network-targeted combination therapies may produce therapeutic benefit with significantly smaller doses of the necessary agents, drugs that were previously considered too toxic at their therapeutically-effective doses might now play a valuable role in the treatment of disease. Thus, this new concept may spawn an enormous new repertoire of molecular-targeted therapeutics for clinical evaluation.
Osmosis is a phenomenon observed in many physical systems that arises when a solute, such as the salt sodium chloride, is present in a vessel which is delimited by a membrane permeable to water but impermeable to the solute. When such a vessel is initially filled with a high concentration of solute and immersed in a fluid the vessel is found to spontaneously swell "absorbing" fluid from the surroundings until reaching a criticial pressure greater than the surrounding fluid. This pressure is referred to as the osmotic pressure. The flow of water and the generation of pressure by osmotic effects plays an important role in many biological systems. On the macroscopic length scale osmotic effects are important in the exchange of nutrients and wastes between the blood and organs. On the microscopic length scale cells and organelles must take special measures to mitigate the effects of osmotic pressure to avoid bursting.
In the biological and medical literature the phenomenon of osmosis is often referred to in somewhat vague terms. In this poster we present a mathematical theory to explain the phenomenon of osmosis and discuss two numerical methods for the simulation of microscopic and macroscopic fluid flows associated with osmosis.
To model microscopic systems we present a stochastic numerical method extending the immersed boundary method of Peskin to include the effects of thermal fluctuations. The inclusion of thermal fluctuations in the method introduces non-smoothness in the solutions of the fluid and particle convection equations. If traditional numerical methods such as Runge-Kutta are used, the non-smoothness requires a small time step making the method impractical for many problems of interest. We develop a numerical method that attains a good level of accuracy for large time steps by making an approximation which exploits a separation of time scales between the fastest modes of the fluid and the particle convection. To model macroscopic systems we develop a finite volume method where the solute particles are described by a continuum field and appropriate reaction-diffusion-convection equations.
In conclusion we present a few applications of the methods. On the microscopic length scale we simulate the osmotic pressure arising from an individual solute molecule. On the macroscopic length scale we show how osmotic effects can be used to pump fluid. We then discuss the potential of the methods to model more complex biological systems and how osmotic effects may offer a novel approach in the design of small length scale devices which pump fluid or exert forces without the need for mechanical parts.
A mathematical description of the collective motion of organisms using a density-velocity model is presented. This model consists of a system of nonlinear parabolic equations, a forced Burgers equation for velocity and a diffusion-convection equation for density. The formation of groups with some typical structure for collective motion of organisms is obtained from the local force resulting from the differences between local density levels and a prescribed density level. Whereas gathering of distant groups or organisms into one group is modeled using the nonlocal force. The existence of a global attractor for a one-dimensional density-velocity model is proved by asymptotic analysis to demonstrate different patterns in the attractors for density. These patterns, which can be characterized into groups, correspond to movement of collective organized groups of organisms such as fish schools and bird flocks. The theoretical results are supplemented with numerical results.
Despite huge advances in biology, our understanding of how spatial patterns arise in development is still very crude. The post-genomic challenge in developmental biology is to elucidate the mechanisms underlying this phenomenon. To this end, somitogenesis, one of the most important and well-studied examples of pattern formation in the developing embryo, is becoming a leading candidate in developmental biology for a study that aims to couple findings at a molecular level with those at a cell and tissue level and lends itself openly to investigation from a more theoretical viewpoint.
Somites arise as the result of a complex process that takes places in the early vertebrate embryo: a seemingly uniform field of cells is organised into discrete blocks via a mechanism which is tightly regulated both in space and time. Further differentiation of the cells within these somitic segments leads to the formation of the vertebrae, ribs and other associated features of the vertebrate musculature.
Various experimental results have shown the existence of a wavefront of gene signalling along the vertebrate embryo, which, coupled with a segmentation clock, is able to gate the cells into blocks that will later go on to form somites. We use a signalling based approach, with cues from the wavefront and the segmentation clock, to mathematically model somite formation and show that our model can reproduce the results seen in vivo when progression of the wavefront is disturbed.
In previous work we showed that cultured cortical networks recorded on 60 channel microelectrode arrays display many stable activity patterns, and operate at the critical point where they produce a power law distribution of neuronal avalanche sizes. To examine the implications of the critical point on information transmission and storage, we tuned network simulations through subcritical, critical and supercritical regimes. Our simulations showed that both information transmission and the number of stable activity patterns are maximized simultaneously, and that this occurs only at the critical point. These results suggest that living cortical networks self-organize to the critical point to optimize both information transmission and storage.
Fluorescence resonance energy transfer (FRET) occurs between two fluorescently tagged proteins when electronic excitation energy is transferred from one protein (the donor) to another (the acceptor) over very small (< 10 nm) distances due to their dipole-dipole interaction. According to the single-distance model (Lakowicz, 1999), FRET efficiency is proportional to the inverse sixth power of the distance between the proteins. More generally within cell membranes, however, FRET efficiency also depends upon stochastic factors including the number of acceptor molecules near the donor (acceptor competition) and the possibility that the acceptor is already excited (acceptor saturation). A sophisticated model for FRET is required to assess the effects of these stochastic factors in a heterogeneous flourophore population.
This poster presents a discrete, stochastic model which explicitly accounts for fluorophore positions, excitation, decay and resonance transfer as stochastic processes. Explicit treatment of molecules and FRET makes this model very intuitive and easy to modify by incorporating different cluster sizes, monomer fractions, and donor-acceptor ratios. We find that FRET efficiencies are very sensitive to inter-cluster distance if there is a monomer fraction. Also, FRET at the edge of clusters is different from FRET in the middle of clusters since the fluorophores have a different number of immediate neighbors. Thus, FRET efficiency depends sensitively upon the typical size, density and surface area of clusters. These measurements are greatly complicated by cluster heterogeneity and would be difficult to predict (and quantify) without a spatially explicit model.
Foot-and-Mouth disease (FMD) is a highly infectious illness of livestock and a serious economic threat. We model the 2001 FMD epidemic in Uruguay using an explicit discrete spatial epidemic model (comprising a series of coupled differential equations) that includes geo-referenced data (i.e. euclidean distances between farms, as estimated in relation to distances between county centroides). The value of spatially explicit models in the development and testing of FMD control measures is tested using the corresponding spatially homogeneous model as basis for comparison. The limitations of spatially homogeneous models are illustrated by their inability to capture effectively observed patterns of spread. For the situation of Uruguay, our discrete spatial model captures a double peak in the epidemic, pattern not observed under the spatially homogeneous model. We define internal (within counties) and external (across counties) reproductive numbers, that is, within and across-county! contributions to the average number of secondary infections under low levels of local infection. We estimate a mean internal Ro= 280.47 while the external Ro= 2.64. Movement restrictions reduced them to Ro_in = 87 and Ro_out = 0.82. Twelve days after the start of the mass vaccination policy the internal reproductive number dropped to less than one. We explore the expected impact of how quickly mass vaccination is implemented after the start of an outbreak. Our model predicts that if the mass vaccination program had been delayed an additional five days, then there would have been 50% more cases. If the vaccination program had been implemented 5 days prior to the actual date, our model predicts the epidemic would have been reduced by 37%.
Patterns of spread for diffusion of ideas and rumors are similar to epidemics. We propose various epidemic-based caricature models for "social contagion" processes including: rumors and ideas. Some of these processes have an inherently efficient rate of initial growth; by using mean-field models one can verify that the average number of secondary rumor spreaders is always favorable to the diffusion of a rumor. In addition, we model rumor spread among populations which support strong fluctuations in density and explore whether its dynamics are independent of the total population dynamics. Furthermore, we propose a model for idea adoption, which supports a Hopf- bifurcation. We also estimate uncertainty in model parameters by fitting mean-field models to data (via stochastic ensemble search) about how Feynman diagrams spread among three theoretical physics communities.
I would like to present a mathematical model for ameboid cell motility. The model is developed using a spring-dashpot system through Newtonian dynamics. It is based on the facts that the cytoskeleton plays a primary role for cell motility and that the cytoplasm is viscoelastic.
Based on the model, the inverse problem can be posed: if a structure like a spring--dashpot system is embedded into the living cell, what kind of characteristic properties must the structure have in order to reproduce a given movement of the cell? This inverse problem is the primary topic of this work.
On one side the model mimics some features of the movement, and on the other side, the solution to the inverse problem provides model parameters that give some insight, principally into the mechanical aspect, but also, through qualitative reasoning, into chemical and biophysical aspects of the cell. Moreover, this analysis can be done locally or globally and in different media by using the simplest possible information: positions of the cell and nuclear membranes.
It is shown that the model and solution to the inverse problem for simulated data sets are highly accurate. An application to a set of live cell imaging data obtained from human brain tumor cells (U87-MG human glioblastoma cell line) then provides an example of the efficiency of the model, through the solution of its inverse problem, as a way of understanding experimental data.
All species modify their abiotic environments to some extent through non-trophic interactions. Ecosystem engineers are species where the magnitude of this modification is sufficiently large that it needs to be accounted for in making predictions about population and community dynamics. We present a simple model of an ecosystem engineer that alters density-dependent or density-independent functions controlling its own population growth. Environmental conditions are altered at a rate dependent on engineer density. These alterations are countered by an exponential decay back to original conditions. Phase plane analysis reveals behavior such as run-away growth and multiple basins of attraction.
Solutions to many chemical reaction networks seem to converge to equilibria, yet proofs of this observation are not widely available. A particular class of systems having the poperty that almost all solutions converge is the class of (strongly) monotone dynamical systems. Assuming that reaction rates are monotone functions of the concentrations of the reagentia, we address the question which chemical reaction networks can be transformed to monotone systems- after taking conservation laws into account. It turns out that a simple grahpical test provides an answer to this question.
This is joint work with David Angeli of the University of Firenze.
Determining the long-time behavior of large dynamical systems has proved to be a remarkably difficult problem. And yet the robustness and stability of molecular networks in biology seems to indicate a certain underlying structure that doesn't change under (some) small changes in the topology or the parameter values. Using the theory of monotone systems, our research group has tried to underline some of the relevant stability features of certain potentially high-dimensional systems. In this talk, I give sufficient qualitative and quantitative conditions for global attractivity and multistability, even for systems that are not monotone themselves, with applications to delay differential equations arising in molecular biology.
The mucus of a human lung consists of long strands of proteins (mucin) in a liquid, thus forming a visco-elastic fluid. As a result, the diffusion of relatively large particles, such as bacteria, does not follow the standard Langevin equations. We investigate a generalized form of the Langevin equations. We then fit experimental observations of beads in mucus to these stochastic differential equation.
A complex vertical canopy structure is common to rainforests around the world and provides geometrical explanations for their high diversity. Light microenvironments allow stratified distributions of organisms. Here I provide quantitative evidence for a well-defined characteristic scale of rainforest layering: a vertical scale defined by a critical tree-height at which canopy gaps percolate, below which a connected landscape of microenvironments emerges. These microenvironments can only give rise to fragmented habitat landscapes. A simple metapopulation model has been provided that identifies characteristic foraging strategies above and below both vertical layers.
We used a simple mathematical model of rat thick ascending limb (TAL) of the loop of Henle to investigate the impact of spatially inhomogeneous NaCl transport and tubular radius (as reported in the experimental literature) on luminal NaCl concentration when sustained, sinusoidal perturbations were superimposed on the steady-state TAL flow. A mathematical model previously devised by us that used homogeneous transport and homogeneous radius predicted that such perturbations result in TAL concentration profiles that are standing waves and that nodes in NaCl concentration occur at the end of the TAL when TAL fluid transit time equals the period of the perturbation. The results for the inhomogeneous case predict that a nodal structure is preserved, but the nodes are transformed into local minima in amplitude, as a function of frequency, and the high frequency response is attenuated. %The results presented in the present study may contribute to gaining a better understanding of tubular flow oscillations that are mediated by tubuloglomerular feedback. In summary, our model predicts that the homogeneous model is robust\ , in the sense that the results for the homogeneous model are essentially preserved in the inhomogeneous case. %HN: the last paragraph - sounds like discussion; probably need, instead, more summary of results. As a consequence of the results presented in this study, the nodal structure of the TAL frequency response curve which illustrates the nonlinear character of the TAL filter is preserved for the generalized case. The new simulations for our model provide results with more attenuated high-frequencies perturbations and distorted waveform. Rather than radius variation, the non-constant transport rate through the tubule seems to be the main factor of influence for these changes.
We employed a recently introduced database approach using a brute-force search of parameter space (Prinz et al., J. Neuropysiol., 90:3998-4015, 2003). The parameter dependence of a multi-compartmental globus pallidus (GP) neuron model is examined using this approach. Our model is built with the GENESIS software, and contains 512 compartments derived from a morphological reconstruction, for each of which 9 voltage-gated Hugdkin-Huxley conductances are specified based on experimental data. A parameter-measure database approach is advantageous in a model as complex as this, since analytical solutions for model behavior are no longer obtainable. The parameters in this study are the densities of the conductances of the model. Parameters were first manually tuned to produce a fit of model behavior with physiological recordings, indicating spontaneous spiking and linear spike frequency responses to current injection pulses (CIP). A brute-force parameter search was then used to examine changes in model behavior for all combinations of 3 density levels of the 8 most significant parameters. The three levels varied for each parameter and were chosen as factors of 2 or 5 times the original value. A total of 19,683 parameter combinations were modeled with the addition of a separate simulation for positive and negative current injections. Each neuron model behavior was recorded in a DB using 34 measures, such as: the spontaneous and CIP period spike rate and spike shape characteristics, the sag amplitude for negative current pulses, and the potential during current pulses and the recovery period. We developed reusable Matlab tools that analyzed the response distributions across parameter settings from the DB. For the wide variety of spike shapes found in the DB, specific methods for identifying action potentials (APs) and finding the AP threshold voltage were necessary.
We found that 50% of the parameter combinations lead to neuron models with spontaneous spiking as the original model. The mean firing rate showed a smooth unimodal distribution with a mean of 8 Hz, indicating a robust behavior of the original parameter combination. The DB also yielded interesting predictions about the influence of individual conductances and dependencies between multiple specific conductance types on model behavior. Parameter sensitivities of a certain measure, and measures that are maximally sensitive to a parameter can be identified from the DB. This is done by observing the relationship between parameter values and measure variances in the DB. These predictions of channel functional roles can be experimentally tested using agonists and blockers with whole cell recording in vitro.
More advanced queries can be formed to analyze constrained subsets of the DB. For instance, consider models with APs with at most 2 ms wide at half-maximal potential. This subset can further be divided into two parts for spontaneous firing rates below and above 40 Hz. These subsets can then be compared to find distinctive parameters and other common measures. Subsets of the database can also be identified by finding the closest match to physiological data. This may allow finding multiple parameter combinations that give realistic behavior. Higher order analysis methods such as principal component analysis, factor analysis, and clustering may be employed on the DB to find key players in defining the model's behavior.
Supported by NINDS.R01 NS039852.
Hemostasis is the normal physiological response to blood vessel injury and is essential to maintaining the integrity of the vascular system. Hemostasis consists of two interacting processes: platelet aggregation and coagulation. The first involves cell-cell adhesion resulting in a platelet aggregate, and the second is an enzyme network that leads to the formation of a fibrin gel. Though both processes contribute to the formation to blood clots, those formed at high shear rates are composed primarily of platelets and clots formed at low shear rates are composed primarily of fibrin gel. In order to understand this phenomenon, a simple model of chemically-induced monomer formation, polymerization, and gelation under shear flow is presented. The model is used to explore how the shear rate and other parameters control the formation of fibrin gel.
We illustrate the interplay between discrete mathematics and molecular biology with results arising from our work on the combinatorics of DNA and RNA molecules. In the absence of complementary strands, single-stranded nucleotide sequences may hybridize with themselves or each other. For the short oligonucleotides or "DNA code words" found in many biomedical applications such hybridizations prevent the DNA segments from performing their desired functions. In contrast, the self-bonding or "secondary structure" of organic RNA molecules is an essential part of their overall structure and function. The discrete nature of the nucleotide base pairing leads naturally to a combinatorial formulation of the question: how is biological information encoded by the selective base pair hybridization of single-stranded DNA and RNA molecules? In this context, we give new combinatorial theorems which yield insight into the folding of RNA molecules and discuss the biochemical properties of random de Bruijn sequences.
A two-dimensional cochlear model is presented, which couples the classical second order partial differential equations (PDEs) of basilar membrane (BM) with a discrete feed-forward outer hair cell (OHC) model for enhanced sensitivity. The enhancement (gain) factor in the model depends on BM displacement in a nonlinear nonlocal manner in order to capture multi-frequency sound interactions and compression effects in a time dependent simulation. The feed-forward mechanism is based on the longitudinal tilt of the OHCs in feeding the mechanical energy onto the BM. A boundary integral method of second order accuracy in space and time is formulated. Though the nonlinear coupling with OHC created an implicit algebraic problem at each time step, the structure of the feed-forward mass matrix is found to permit a decomposition into a sum of a time independent symmetric positive definite part and the remaining time dependent part. A fast iterative method is devised and shown to converge with only the inversion of the time independent part of the mass matrix. The time dependent computation is studied by comparing with steady state solutions (frequency domain solutions) in the linear regime, and by a convergence study in the nonlinear regime. Results are shown on OHC amplification of BM responses, compressive output for large intensity input, and nonlinear multi-tone interactions such as tonal suppressions and distortion products. Qualitative agreement with experimental data is observed.
Many immune responses are initiated through the binding of antibody receptors present on the surface of B cells. Once activated, these B cells proliferate and undergo a process of somatic hypermutation whereby point mutations are introduced into the DNA coding for their antibody receptor. These mutations have important consequences for the affinity and specificity of the response.
Despite the significance of somatic hypermutation, precise estimates of the mutation rate do not currently exist. We have used mathematical modeling and computer simulation to interpret data from microdissection studies and estimate this rate more accurately than previously possible. Each microdissection experiment provides a number of clonally related sequences that, through the analysis of shared mutations, can be genealogically related to each other. The 'shape' of these clonal trees is influenced by many processes including the hypermutation rate. Modeling and simulation are used to relate the 'shapes' of these trees to the underlying biology. We have developed two different methods to estimate the mutation rate based on these data. The first relies on a computer simulation of clonal expansion as a stochastic branching process. The second is analytical. By combining these models with optimization techniques we have produced precise methods for estimating the mutation rate in vivo.
The results of applying these methods to experimental data from autoimmune mice were used to demonstrate the existence of hypermutating B cells in an unexpected location in the spleen. This has great implications for understanding how autoimmune diseases such as Lupus begin.
I will present the use qualitative network models to answer whether neural progenitor cell fate decisions, specifically their timing, can be explained by viewing DNA damage response as an intrinsic developmental timer. We rely on cell-population scale data, including mosaic composition with respect to aneuploidy, in our attempt to validate qualitative network models that encompass modularized aspects of DNA damage response, and that integrate the molecular pathways involved in proliferation, differentiation, and apoptosis. Of course, such modeling, in attempting to approach cell cycle and cell fate control, is far from completed, and we will briefly discuss the mathematical challenges that lie ahead.
Synaptic facilitation (SF) is a form of a transient increase in synaptic strength elicited with just one or several stimulation pulses, and decaying with time constants from tens to hundreds of milliseconds. At some synapses, SF may be caused by the increase in the activity-induced Ca2+ influx; however, in many other synaptic types SF is believed to result from the presynaptic accumulation of residual Ca2+, under conditions of constant Ca2+ current from one pulse to the next. In the latter case, it is not known whether it is free or bound residual Ca2+ that underlies SF. Experimental work has demonstrated that an increase in intrasynaptic Ca2+ buffering capacity leads to a rapid reduction in both the baseline synaptic response and in the magnitude of SF; this is often viewed as a proof that SF is caused by the accumulation of Ca2+ in free form. In the past we have explored two variations of such free residual Ca2+ hypothesis of SF: the so-called two-site model, and the buffer saturation mechanism. However, here we show that a model including the contribution of bound Ca2+ to SF is also consistent with the observed effect of exogenous Ca2+ buffers on synaptic response, and thus represents a viable alternative to the two-site free residual Ca2+ model. In particular, we show that such hybrid free/bound Ca2+ model is not in contradiction with the Kamiya-Zucker protocol (1994, Nature 371:603), whereby the synaptic strength of the crayfish neuromuscular junction is seen to decrease within few milliseconds of a UV-flash photolysis liberating a Ca2+ buffering compound. While evidence indicates that buffer saturation may underlie SF at calbindin-positive central synapses (Blatow et al, 2003, Neuron 38:79), we conclude that SF at other synaptic types may well involve a slow Ca2+ unbinding step from a putative Ca2+ release sensor.
Actin polymerization is currently identified as the dominant mechanism responsible for the propulsion of certain pathogenic bacteria, viruses, endosomes, and endogenous vesicles in eukaryotic cels. Although the details of the underlying biochemistry associated with the phenomenon of actin-based propulsion are understood, the exact biophysical mechanism by which force is generated is still a matter of debate. Mathematical models have been proposed and investigated in both the microscopic and the mesoscopic scale. These include the tethered ratchet model by Mogilner and Oster, which provides a very interesting scenario of force generation by individual filaments, and the elastic-propulsion model by Gerbal et al., which focuses on the elastic forces exerted by the actin tail at the mesoscopic level. In this talk we discuss a new multi-scale approach to the phenomenon of actin-based motility which aims to bridge the gap between the two different scales represented in these models. The proposed model is applied in a variety of experimental settings at the mesoscopic level, and leads to the reduction of complicated molecular scenarios at the microscopic level to observable mesoscopic dynamics, which can be tested experimentally.
Differential equation models of chemical mechanisms are considered. The chemical mechanism is represented by a bipartite graph. The possibility for a positive equilibrium point to become unstable for some set of parameters is connected to the cycle structure of the bipartite graph. Necessary conditions for multistability or oscillations are derived based on the graph structure. The advantage of using the bipartite graph, compared to the interaction graph, will be discussed. This work is related to the concept of positive and negative feedback cycles and to the graph theoretic approach of studying chemical mechanisms. The same theory is useful when diffusion or delays are introduced in the model. Examples from ordinary differential, reaction-diffusion and delay differential equations will be given.
Despite the fact that many plant communities are diverse, the study of plant competition for light continues to be shrouded in mystery. We argue that the vertical placement of a species' leaves relative to others in the stand can promote coexistence. We do so by formulating and analyzing a mathematical model of plant competition in which the two populations interact only by their shared use of the light. The model's assumptions give rise to a Kolmogorov system of integro-differential equations. We then use implicit methods to show that the species' nullclines, which are defined only implicitly, can intersect at most once, and that when they do intersect, coexistence is always stable. We conclude that the model does not exhibit founder control, and so we can divide parameter space into regions in which one of the various outcomes of competition occur.
Information rapidly transfers between neurons at malleable connections called synapses. Coincident signals of voltage and neurotransmitter must be internalized by the cell to adapt its connections. Calcium released from internal stores and triggered by inositol triphosphate (IP3) and calcium itself plays the messenger. The endoplasmic reticulum (ER) is a contiguous intracellular store for calcium. The dynamics of IP3 and calcium induced calcium release (ICICR) through the IP3 receptor (IP3R) is balanced by receptor inactivation and calcium reuptake into the ER or extracellular extrusion. The location of slow (relative to voltage) regenerative calcium efflux and of subsequent system recovery travels bidirectionally from an initiating site. After reaching the nucleus, calcium induces a cascade which upregulates synaptic machinery.
To explain the initiation point(s) of calcium waves we use a two variable ICICR model in which cytosolic calcium diffuses. Inhomogeneity in IP3 concentration and IP3R density can account for these foci. Boundaries in parameter space of IP3 and IP3R levels separate regions of no calcium, uniform calcium release, and calcium waves. Wave speed dependence on IP3 and IP3R densities are estimated. Results correspond with experimental findings in rat pyramidal neurons.
Genetic algorithms have been increasingly used in phylogenetics during the last ten years to find the optimal phylogenetic tree/s from the space of all possible leaf-labeled topologies. This combinatorial optimization problem (phylogenetic tree search) is compute bound and must be approached through heuristics for large and biologically interesting datasets. Results on parallel implementations of genetic algorithms are presented here for the analysis of datasets with large number of species (between several hundreds and several thousands) using Beowulf clusters.
In non-excitable cells agonist stimulation can induce, via production of inositol-1,4,5-trisphosphate (IP3), transient increases in cytoplasmic calcium so called calcium oscillations. Several mechanisms contribute to the generation and termination of these transients (e.g. calcium induced calcium release, calcium sequestration, inactivation of the IP3 receptors).
I will present a model for the IP3 metabolism. This model includes the IP3 precursors (phosphoinositol, -phosphate, and -bisphosphate). The model parameters could be fitted to experimental data. The model of the IP3 metabolism is combined with a model for the Ca2+ dynamics that includes the main calcium transport processes and the dynamics of the IP3 receptors. I will show that it is not possible, just by activating the IP3 producing enzyme PLC to generate a long lasting IP3 signal and so long lasting Ca2+ oscillations. One solution to this problem is when PLC is activated by Ca2+. In this case Ca2+ activation of PLC is a precondition to generate Ca2+ signals.
Most tumors in vivo become highly non-homogeneous even at very early stages of their growth. In order to address different aspects of tumor formation and development on the level of single cells, we propose a two-dimensional time-dependent mathematical model taking explicitly into account individually regulated biomechanical processes of tumor cells and communication between cells and their microenvironment. A mathematical framework of this model is constituted by the immersed boundary method and couples the dynamics of separate elastic cells with the continuous description of a viscous incompressible cytoplasm inside the cells and the extracellular matrix outside the tissue. I will present numerical simulations addressing the self-organized formation of tumor clusters, the development of tumor microregions and the geometric patterns of ductal carcinomas in situ.
This poster concentrates on applying dynamical systems to computational neuroscience. We concentrate on a model for short term synaptic changes within populations of different types of neurons. Such changes are known as short term plasticity. We model the behavior observed in experiments studying epileptic seizures.
We present a mathematical model for the rod phototransduction cascade, describing the diffusion of the second messengers, cGMP and Ca2+ in signaling in the rod outer segment of vertebrates. Numerical simulations of the response of dark-adapted Salamander rods to dim light flashes are performed. The results are consistent with experimental data. The simulations are based on finite element discretization and implemented in Matlab.
The aim of this work is to formulate a mathematical model of rotavirus infection that incorporates vaccination of infants as part of a contingency plan against rotavirus transmission. Better understanding of the transmission mechanics of rotavirus infection and predicting possible future epidemics is of high priority due to the high incidence and morbidity observed in past and concurrent epidemics. Rotavirus is the most common cause of severe diarrhea among children, resulting in the hospitalization of approximately 55,000 children each year in the United States and the death of over 600,000 children annually worldwide. About 95% of all children in the United States are infected by 2 years of age. Recently, a vaccine against rotavirus infection has been approved in Mexico in July 2004. The model presented is formulated on age-structure basis since rotavirus infection mostly occurs to young children under age 2 years old. According to computer simulation experiments wit! h this model, one could devise the most efficient strategies for implementing a mass vaccination program. Also there are seasonal patterns of rotavirus epidemic observed in many geographic areas including US, Australia and Mexico. For instance, timing of rotavirus activity is highly region-sensitive in US - peak activity occurred first in the Southwest from October to December and last on the Northeast in April. The cause of the seasonal differences in rotavirus activity by regions will be studied. We will explore the mathematical interpretation of such pattern and predict possible future epidemics. Our mathematical model can also be used to determine the vaccination priority based on the status of acquired or innate immunity of each age group. Thus one can develop an efficient vaccination policy targeting specific age groups and the timely seasonal implementation of the vaccination program each year. In summary, developing specific vaccination strategies will allow u! s to minimize resources (vaccines, medical personnel) by targeting the right age groups and the timely implementation of such vaccination campaigns against rotavirus infection.
Directed migration of eukaryotic cells is involved in processes such as embryonic development, wound healing, and the metastasis of cancer. The motile machinery of cells involved in these types of processes is controlled by a diffusible signal released into the surrounding tissue from sources that depend on the particular process in question. Some major questions regarding cell motility are as follows: How does a cell exert force at the leading edge allowing it to move? How are the material properties of the cell modulated to allow extension of the leading edge and motility in general? What kind of a role do mechanical stresses that are transmitted to and from the cell play? To begin answering these questions, I focus on the role of mechanics in amoeboid cell motility and present a three-dimensional phenomenological model of cellular mechanics in which the protrusion and retraction typically associated with cell 'crawling' are incorporated through a localized multiplicative decomposition of the deformation gradient into active and passive parts. The active part of the deformation gradient incorporates the role of actin dynamics in cell motility. Experiments have shown that the passive response of the cell is viscoelastic, and this rheology is incorporated through an appropriately chosen constitutive equation. Finite element numerical simulations will be shown, and I will also discuss numerical issues associated with solving this nonlinear model.
Using a large-scale modle of the Macaque Primary Visual Cortex (V1), we offer an explanation for the observd ensemble activity in local field potential (LFP) measurements. Our network model represents the activity of a patch of layer 4Calpha of V1, reproduces many of the single unit observations, covering both extra- and intra-cellular measurements, and accounts for spatial summation porperties through a balance between feedforward excitation and recurrent cortical connections. Analysis of the model reveals features consistent with LFP measurements of Henrie & Shapley: Ensembles of model neurons display gamma-band modulations which become more apparent as ensemble size is increased. Furthermore, the network model shows diverse steady-state orientation selectivity as measured with drifting sinusoidal gratings in a manner consistent with spectral analysis of the LFP in V1. In particular, some recording sites, at which very orientation selective neurons are found, also have qui! te selective ensemble responses. Other sites show non-selective ensemble response even when single neurons at the same recording site are highly selective. In the model, these two cases correspond to lcoations near "iso-orientation" domains and "pinwheel centers" respectively and can be attributed to differing effective lengthscales of cortical coupling in angular coordinates.
Dendrites form the major components of neurons. They are complex, branching structures that receive and process thousands of synaptic inputs from other neurons. The dendrites of many neurons are equipped with excitable channels located in dendritic spines that can support an all-or-nothing action potential response to an excitatory synaptic input. Here we develop a mathematical model of dendrites based upon a generalisation of the analytically tractable Spike-Diffuse-Spike model of dendritic tissue. The active membrane dynamics of spines are treated using an analytically tractable integrate-and-fire process. The spines are connected to a passive dendritic cable at a discrete set of points. The model is computationally inexpensive and ideally suited for the study of neural response to complicated spatio-temporal patterns of synaptic input.
For over 100 years foresters and plant physiologists have known that conduits for water transport in woody plants are smaller in branch tips than in the main stem. This increase in conduit size from top to bottom is thought to reflect a trade-off essential to plant survival: maximize conductance to water transport while minimizing interruptions to continuous flow. Yet, empirical data do not support any theory of whole-plant size-variation. We present the first direct empirical evidence of an ontogenetically stable hydraulic design in woody plants. We also derive a scaling prediction for conduit taper based on a theory of fractal branching networks that optimally decouples hydraulic conductance from path length. We show that the observed vessel tapering within Fraxinus americana (white ash) agrees with the scaling prediction: radii increase with distance from the petiole to the 1/4-power, independent of tree height or age. We discuss the possible physiological basis for this scaling, as well as its relevance to alternative theories of hydraulic design.
In this paper, we present a PDE model to investigate quantitatively the dorsoventral morphogen gradient formation in the embryos of a typical vertebrate: zebrafish. BMP ligands binding with cell receptors act as a morphogen to induce tissue patterning. The enhancement role of inhibitor chordin with the cooperation of the tolloid metalloprotease on the BMP-receptor concentration is analyzed via numerical simulations. Our simulation results are consistent with existing biological experimental observations. Effects of feedback of BMP signals are discussed. This is a joint work with Arthur Lander, Qing Nie and Fred Wan at UC Irvine.