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Second Young Researchers Workshop in Mathematical
Biology
Nuclear Magnetic Resonance (NMR) is a spectroscopic technique that involves the study of molecular structures via the interaction of radio-frequency electromagnetic radiation with the nuclei immersed in a strong magnetic field. NMR experiments provide typically two kinds of structural constraints; distance constraints using solution state NMR and orientational constraints using solid state NMR (ssNMR). Membrane protein structures are extemely hard to determine. ssNMR has been found to be effective in solving transmembrane protein structures. The mathematical problems that arise while solving for the three dimensional atomic structure of a membrane protein from ssNMR data are discussed here. Tools from linear algebra and differential geometry are used.
A new variational region based model for a simultaneous image segmentation and registration is proposed. The purpose of the model is to segment and register novel images simultaneously using a modified Mumford-Shah technique and region intensity values. The segmentation is obtained by minimizing a modified Mumford-Shah model. A global rigid registration is assisted by the segmentation information and region intensity values. In addition, the model can also be applied to the case of non-rigid registration. The numerical experiments of the proposed model are tested against simulated normal noisy human-brain MRI images. The preliminary experimental results show the effectiveness of the model in detecting the boundaries of the given objects and registering novel images simultaneously.
Evidence from microscopy suggests that the spatial organization and heterogeneity of receptors and other signaling proteins is a key feature of many signaling pathways. Yet, the vast majority of mathematical models for cell signaling (ODE-based reaction network models) are based on the assumption that each biochemical species is uniformly distributed throughout the cell. The widespread use of ODEs is in part due to the increased computational cost of spatiotemporal modeling and in part due to the lack of detailed knowledge of the model parameters associated with the passive and active transport mechanisms of the cell. While the spatiotemporal modeling is more challenging than well-mixed modeling, spatiotemporal modeling can provide new insights into microscopic events leading to an improved understanding of signal transduction processes that are important at the molecular level. The work presented here couples kinetic Monte Carlo simulations (used to describe the spatial and temporal dynamics of the ErbB receptor on the cell membrane) with reaction-advection-diffusion systems of PDEs (used to describe downstream ErbB intracellular signaling). The results clearly show that spatial heterogeneity is important for the control and regulation of the ErbB network.
Bifurcation theory and perturbation theory can be combined with a knowledge of the underlying circuitry of the visual cortex to produce an elegant story explaining the phenomenon of visual hallucinations. A key insight is the application of an important set of ideas concerning spontaneous pattern formation introduced by Turing in 1952. The basic mechanism is a diffusion driven linear instability favoring a particular wavelength that determines the size of the ensuing stripe or spot periodicity of the emerging spatial pattern. Competition between short range excitation and longer range inhibition in the connectivity profile of cortical neurons provides the difference in diffusion length scales necessary for the Turing mechanism to occur and has been proven by Ermentrout and Cowan to be sufficient to explain the generation of a subset of geometric hallucinations reported. Incorporating further details of the cortical circuitry, namely that neurons are also weakly co! nnected to other neurons sharing a particular stimulus orientation or spatial frequency preference at very long ranges and the resulting shift-twist symmetry of the neuronal connectivity, improves the story. We expand this approach in order to be able to include the tuned responses of cortical neurons to additional visual stimulus features such as motion, color and disparity. We apply a study of nonlinear dynamics similar to the analysis of wave propagation in a crystal lattice to demonstrate how the spatial pattern formed through the Turing instability can be pinned to the geometric layout of various feature preferences. The perturbation analysis is analogous to solving the Schrödinger equation in a weak periodic potential. Competition between the local isotropic connections which produce patterns of activity via the Turing mechanism and the weaker patchy lateral connections that depend on a neuron's particular set of feature preferences create long wavelength affects analo! gous to commensurate-incommensurate transitions found in fluid systems under a spatially periodic driving force. In this way we hope to better understand how the intrinsic architecture of the visual cortex can generate patterns of activity that underlie visual hallucinations.
The challenge for drug design is to create molecules with optimal function that also partition into appropriate in vivo compartment(s). Pharmacokinetics can be improved by ancillary molecules, such as cyclodextrins, tha increase the effective concentrations of hydrophyic drugs in the blood by encapsulating the drug. However, many drug targets are located inside cells and encapsulating molecules have not been developed that would increase the effective concentration of drugs inside cells. Here we propose that RNA aptamers might preform the same functions inside cells, but with higher specificity, as do cyclodextrins in the body fluids.
Existing ecological theory suggests that environmental noise should increase the amount of variation observed in the size of populations across time. In contrast, two recent empirical studies suggest that populations may exhibit less variation in a more variable environment. Here I present theory illustrating 4 mechanisms that could explain this result. Environmental variation can reduce the amplitude of cycles due to 1) non-linear averaging, 2) counter-acting other environmental variation, 3) destructive interference, or 4) compensatory dynamics at a tropic level. In real systems, these mechanisms may act in separately or in concert.
Plants have two main modes of pollen dispersal; via wind or by animal. To date, both of these mechanisms have been examined using the same diffusive models. In order to better understand the subtleties of these dispersal modes, a cellular automata model is used to predict pollen range, genetic diversity, and genetic differentiation. Among the main controlling factors in the model are propensity on the pollen movement, the probability of pollination, and the density of individuals of the species.
We present an automated spectral interpretation algorithm that determines peptide sequence identity directly from MS tandem mass spectra (de novo sequencing). This technology and its ability to perform de novo sequence identification, independent of a genome sequence or annotation, will be of critical importance for localization of single nucleotide polymorphisms, identification and characterization of translational editing, and the detection of genes where either genomic sequences are not available or their annotation is incorrect.
Data discretization, also known as binning, is a frequently used technique in computer science, statistics, and their applications to biological data analysis. A new discretization method is presented which was motivated by a new reverse engineering approach to modeling the biochemical network governing the cellular response of yeast under oxidative stress. While the method can be used for data clustering, it was developed to answer the need for a discretization algorithm that is specifically designed to handle typical systems biology data: a large number of short multivariate time series of heterogeneous data, such as transcript, protein, and metabolite concentration measurements. To such data sets, statistical discretization methods are hardly applicable due to the prohibitive cost of obtaining sample sets of sufficient size. The method transforms real-valued data into a finite number of discrete values. Novel aspects are the incorporation of an information-theoretic criterion and a criterion to determine the optimal number of values. As several modeling techniques for biochemical networks employ discrete variable states, the method needs to preserve the dynamic features of the time series as well as be robust to noise in the experimental data. An example demonstrating the method's qualities will be shown.
We consider neuronal models where the transition from resting behavior to spiking behavior takes place via a Hopf Bifurcation. These models can exhibit resonance, where a certain frequency of input elicits the largest response in the cell. We study the case where the synaptic input to such a cell undergoes both facilitation and depression. The result is a band pass filter where the synapse is strongest at a certain driving frequency. This is known as synaptic resonance. We look for driving frequencies for the synapse that result in the maximum output in the resonant cell.
We consider the directed evolution of a population after an intervention that has significantly altered the underlying fitness landscape. We model the space of genotypes as a distributive lattice; the fitness landscape is a real-valued function on that lattice. The risk of escape from intervention, i.e., the probability that the population develops an escape mutant before extinction, is encoded in the risk polynomial. Tools from algebraic combinatorics are applied to compute the risk polynomial in terms of the fitness landscape. In an application to the development of drug resistance in HIV, we study the risk of viral escape from treatment with the protease inhibitors ritonavir and indinavir.
The introduction of delays into ordinary differential equation models of populations complicates the dynamics considerably. Often, globally attractive equilibria become unstable, and periodic solutions emerge. I study this scenario in one and two dimensional population and predator-prey models.
Prediction of protein residue-residue contacts from sequence information is an important and difficult problem in structural bioinformatics. Most methods use correlated mutation analysis to detect such contacts. We developed a new approach (P2PConPred) for intra-protein contact prediction, which is oriented for detecting direct physical contacts. Our method uses a novel pair-to-pair substitution matrix (P2PMAT) derived from accurate protein multiple sequence alignments with available representative structures. The P2PMAT matrix integrates the probabilities of contacting and non-contacting protein sites. Incorporating evolutionary (sequence) conservation of residues with information regarding correlated substitutions is natural and effective in our P2PMAT matrix. Our P2PConPred method is most sensitive for contact prediction in the protein cores. Core residues are effectively identified from sequence information alone by means of a predicted surface accessibility of! proteins. Our method improves protein core contact prediction by 1.25 and 1.6 fold over the contact prediction method of Gobel et. al. and that of Singer et. al., respectively. Combining our approach with other approaches for calculating correlated mutations is expected to be beneficial. The basic approach we developed can also be naturally applied to protein structure prediction and de-novo drug design.
The Anchor Cell (AC) and the Ventral Uterine Precursor Cell (VU) participate in a critical interaction during the development of the C.elegans hermaphrodite gonad. We propose a new mechanism for this interaction and show, via analysis and computation of our mathematical model, that this mechanism resolves outstanding questions regarding the AC/VU fate decision.
Cardiac mechanics can be modeled as the dynamic interaction of
a viscous incompressible fluid (the blood) and a (visco-)elastic
structure (the muscular walls and the valves of the heart). The
immersed boundary (IB) method is a mathematical formulation and
numerical approach to such problems. In this presentation, we describe
an adaptive version of the IB method introduced in [1,2]. The adaptive
scheme employs the same hierarchical structured grid approach (but
a different numerical scheme) as the two-dimensional adaptive IB
method of Roma et al. [3,4] and is based on a formally second order
accurate (i.e., second order accurate for problems with sufficiently
smooth solutions) version of the IB method that we have recently
described [5]. We present results obtained from the application
of this adaptive methodology to the three-dimensional simulation
of blood flow in the heart. The results obtained by the adaptive
method show good qualitative agreement with simu! lation results
obtained by earlier non-adaptive versions of the method, but the
flow in the vicinity of the model heart valves indicates that the
new methodology provides dramatically enhanced boundary layer resolution.
Differences are also observed in the details of the flow about the
mitral valve leaflets.
Stoichiometric Network Theory is a constriants-based, optimization approach for quantitative analysis of the phenotypes of large-scale biochemical networks while avoiding the use of detailed kinetics. This approach uses the reaction stoichiometric matrix in conjunction with constraints provided by flux balance and energy balance to guarantee mass conserved and thermodynamically allowable predictions, respectively. However, to date, the flux and energy balance constraints have not been implemented simultaneously because optimization under the combined constraints is nonlinear. I will present a sequential quadratic programming algorithm that solves the nonlinear optimization problem. A simple example and the system of fermentation in Saccharomyces cerevisiae are used to illustrate the new method. The algorithm allows the use of nonlinear objective functions. As a result, a novel optimization with respect to the heat dissipation rate of a system can be suggested. The importance of incorporating interactions between a model network and its surroundings is also emphasized.
Software which is readily available to the neuroscience community can be used to reconstruct cortical surfaces from magnetic resonance imaging (MRI) data. Topologically correct cortical surfaces representing a white matter (WM) surface (which occurs at the white matter/gray matter interface) and a gray matter (GM) surface (which occurs at the gray matter/cerebrospinal fluid interface) can be created. Using these surfaces, we present an approach for producing a cortical mantle volume representing the gray matter. We then use this cortical mantle volume to restrict the analysis of functional MRI (fMRI) data to the gray matter. These results are then compared to a similar analysis using the entire masked volume.
The medial entorhinal cortex (mEC) participates in theta (4-12Hz), beta (15-25Hz), and gamma (30-80Hz) frequency rhythms, which are thought to play an important role in the formation of neuronal ensembles. In vivo studies have shown theta-nested gamma oscillations in the superficial layers of the EC during exploratory behavior and REM sleep, while in-vitro studies have shown theta modulation of gamma activity during kainate application. In vivo, the theta component is thought to arise mainly due to inputs from the medial septum. Recent findings from slice preparations have shown several rhythmic activity patterns in the superficial layers of the entorhinal cortex during the presence and absence of kainate and NMDAR antagonists. Under kainate, NMDAR block causes a decrease in gamma activity, while in the absence of kainate gamma activity increases. We investigate the mechanisms for these surprising experimental results with a model consisting of populations of pyra! midal cells, fast-spiking interneurons, theta-producing interneurons, and stellate cells. In addition, we investigate rhythms generated by the addition of septal input to the model.
In this study, we combine insights from human macroscopic experimental measures and computational neural network modeling to test the hypothesis that perception correlates with cortical activity measured in both the time and frequency domain, and that this activity is mediated by specific cellular level neuronal events. This is accomplished using a two-fold approach. First, we experimentally probe the influence of perception on human cortical rhythms using a somatosensory tactile detection task. Specifically, we will use techniques recently developed at the Arthinoula A. Martinos Center for Biomedical Imaging to measure magnetoencephalography (MEG) signals during presentation of a threshold level somatosensory stimulus. We analyze the signals generated in the primary somatosensory cortex (SI). In the time domain, we will measure amplitudes and latencies of evoked responses, and in the frequency domain will measure spectral power and phase-locking. We will compare ! these measures during perceived and unperceived trials. Second, we use biophysically based neural network modeling to test if changes in the level of " top down" modulation create a systematic biophysical link between changes in time and frequency domain activity, which are consistent with the experimental results. This approach entails the development of a model of a laminated cortical column(s) that reproduces the oscillatory current dipoles that are measured extracranially with MEG. This two-fold approach may lead to a better understanding of the macroscopic and cellular mechanisms of perception.
Cell motility is essential to cancer invasion. Cells use a variety
of directed migration mechanisms in response to chemokinesis (chemotaxis),
rigidity (mechanotaxis), or extracellular matrix gradients (haptotaxis).
Many cancer studies focused on chemotaxis as the main event for
a cell to invade, however, little is known on the role of haptotaxis
in cancer invasion. Our goal is to produce a quantitative understanding
of the haptotatic component of cancer invasion by using a mathematical
model. To this end, our first step is to implement original techniques
for measuring physical parameters of haptotaxis under controlled
conditions. We modified a methodology developed at Vanderbilt to
study chemotaxis (Walker et al., 2005). Instead of creating a chemokine
gradient, we used a gradient mixer to lay down a gradient of fluorescently-labeled
extracellular matrix protein. Then, cells were introduced into the
device and allowed to adhere and spread. Haptotaxis was followed
! by videomicroscopy. Fluorescent pictures were acquired to assess
the slope of the gradient. Brightfield movies were analyzed to calculate
parameters such as cell speed and trajectory.
Excitatory-inhibitory networks arise in many neuronal systems.
Examples include models for thalamic sleep rhythms and parkinsonian
tremor. Such networks have been shown to exhibit a rich structure
of firing patterns, including synchronous activity, irregular and
chaotic dynamics and propagating wave-like behavior. Computational
and analytical methods have been employed extensively to understand
those patterns in networks with one spatial dimension. However there
has been very little work devoted to the numerical or analytical
investigation of higher dimensional networks. We consider two-dimensional
sheets of synaptically coupled excitatory and inhibitory neurons
and explore the types of additional patterns that emerge. The models
consist of large systems of nonlinear differential equations and
represent the interactions between two neural populations: the subthalamic
nucleus and the globus pallidus. The membrane potential in those
models exhibits bursting patterns and thus reveals several time
scales. Using a dynamical systems approach we analyze the mechanisms
underlying such bursts and then reduce this complex high-dimensional
model to a simpler yet biophysically meaningful system.
In the current study, we derive a mathematical model describing
fluorescence recovery after photobleaching (FRAP) of the Golgi complex,
the endoplasmic reticulum (ER) or cytosol in terms of interactions
between different pools of depalmitoylation mutant GFP-HRas proteins
(DMGRs).
Mesencephalic dopamine neurons ordinarily will not fire faster than about 10/s in response to somatic current injections. However, in response to dendritic excitation, much higher rates are briefly attained. In an analysis of a simplified biophysical model, we suggest a way such high-frequency transient firing may be evoked. Our model represents the neuron as a number of electrically coupled compartments with different natural frequencies, which correspond to the soma and parts of the dendrite. We reduce this model, substituting all the diversity of the compartments that describes real dendritic geometry by a pair of compartments: the slowest, somatic and the fastest, the most distal dendritic one. We have shown that, in the absence of any synaptic stimulation, oscillatory pattern in this model is controlled by the somatic compartment, and, therefore, has a very low frequency. We consider and compare NMDA and AMPA activation applied to the dendritic compartment. Our main result is that activation of the dendritic NMDA receptors evokes oscillations at a much higher frequency. We have also shown that dendritic AMPA activation, by contrast, cannot increase the frequency significantly. The major dynamical question left is how the dendritic frequency can dominate during the application of NMDA. We employ a phenomenon of localization to explain this behavior.
D4 dopamine receptors play an important role in neuropsychiatric disorders involving working memory deficits, such as schizophrenia, attention-deficit/hyperactivity disorder (ADHD) and other disorders of cognition. The impaired methylation has also been documented in different mental diseases. Given a hypothesized role of D4-induced PLM in ion channel functioning [1], we explore in a simple cortical network an influence of this modulation on ability of network to synchronize at gamma frequency and on spike train properties. We model how this mechanism can induce spike trains and modulate its duration and rate, which can mimic short-term memory state. Changes in connectivity within the network influence the effect of DA. Different behavioral characteristics of DRD4 knockout mouse model are believed to be indicative of changes in the striatum (STr), nucleus accumbens (NAc), and prefrontal cortex (PFC), all areas known to be smaller in brains of human ADHD patients. The principal cortical-subcortical neural network for model of schizophrenia has also PFC and NAc/Striatum components. We check how the proposed mechanism can affect transmission in such cortical-subcortical neural network. Results of this study should have significance for further consideration of D4R as novel treatment for ADHD and schizophrenia. 1) Deth, R.C., Kuznetsova A. and Waly, M., 2004, "Attention-related Signaling Activities of the D4 Dopamine Receptor" in Cognitive Neuroscience of Attention, Michael Posner Ed., Guilford Publications, p. 269-282.
I use dynamical systems techniques to model the pheromone response
signaling pathway in Saccharomyces cerevisiae yeast. This pathway
in yeast is a prototype of regulatory systems that govern response
to external stimuli in higher eukaryotic (having a nucleus) organisms.
The S. cerevisiae yeast come in two mating types, MATa and MATa.
When a yeast cell is exposed to pheromone from yeast of the opposite
mating type, a sequence of molecular events happen inside the cell.
The components that affect this sequence comprise the pheromone
response signaling system.
The yeast Saccharomyces cerevisiae has been used as a model eukaryote
for fundamental and applied studies, including stress research.
The cellular responses to oxidative stress involve several biological
processes, and follow a complex regulation. Understanding which
signals trigger the response to the oxidant, which processes are
involved, and what is the temporal dynamics of the response is essential.
Oxidative stress is related to processes such as ageing, apoptosis,
cancer, and the information obtained using yeast cells as a model
will certainly help to clarify questions involving higher eukaryotes.
In this work we studied the genome-wide response of S. cerevisiae
to oxidative stress induced by cumene hydroperoxide (CHP). The changes
in gene expression were monitored at the transcriptional level,
from a dynamical point of view, spanning a time range of 3 to 120
min after the addition of the oxidant. Affymetrix Yeast
Genome S98 arrays were used for this purpose.
Many of the smallest flying insects clap their wings together at the end of each upstroke and fling them apart at the end of each downstroke to augment the lift forces generated during flight. Previous work using rigid wings has shown that at low Reynolds numbers, this mechanism is rather inefficient as large drag forces are produced when the wings are clapped together and pulled apart. In this paper, the immersed boundary was used method to investigate whether or not wing flexibility improves aerodynamic performance during low Reynolds number 'clap and fling.' Our results suggest, for a certain range of flexibilities, that wings with rigid leading edges and flexible trailing edges yield improved aerodynamic performance relative to the rigid wing case. Different wing designs are also shown to yield improved lift generation or improved aerodynamic efficiency.
A bipartite graph, used to represent the chemical mechanism connects its structure with the dynamic properties of the corresponding differential equation model. If certain subgraphs are present in the graph then the conventional model can admit instabilities for some values of the system's parameters. In models with delay or diffusion an instability is usually associated with oscillations or pattern formation respectively. Using the same bipartite graph, the diffusion or delay instabilities can be characterized also in terms of the structure of the chemical mechanism.
Molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy, by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps on the cytoskeleton, and carrying a molecular load. With sufficient resolution, single molecule recordings of motor steps appear as a stochastic sequence of dwells, resembling a staircase. Here, we developed maximum likelihood algorithms that separate the dwell time sequence from noise, estimate the mean and variance of the step size between consecutive substrate positions, and estimate the rate constants of conformation and step transitions of the molecular motor. We model the motor with an infinite but periodic Markov model, reduced to a small model reflecting the periodic chemistry of each step. The dwell sequence is extracted using maximum likelihood dynamic programming algorithms. The kinetics are estimated from the dwell time sequence by numerical maximization of the likelihood fun! ction for discrete time Markov models. The algorithm can fit models with arbitrary chemistry and allows global fitting across stationary and nonstationary experimental conditions, and user-defined constraints on rate constants. Our results show that when the measurement noise is within the nanometer range, steps as small as 8 nm can be analyzed. The algorithm is implemented in the free QuB software (www.qub.buffalo.edu).
In order to study how the convergence of many variable neurons on a single target can sharpen timing information, we investigate the limit as the number of input neurons and the number of incoming spikes required to fire the target both get large with the ratio fixed. We prove that the standard deviation of the firing time of the target cell goes to zero in this limit and we derive the asymptotic forms of the density and the standard deviation near the limit. We use the theorems to understand the behavior of octopus cells in the mammalian cochlear nucleus.
In this paper, classical epidemiology models are modified to incorporate
the effects of control stages. Effects of isolation and qua-ranting
the traced contacts to wipe out the disease from the population
are highlighted. Three models with different contact tracing functions
for new emerging diseases are described and studied. A model with
general contact tracing rate is also framed. Threshold, disease
free equilibrium and its global stability criteria for the models
are explored.
Morphogen gradients of Wingless are involved in patterning the Drosophila embryo and imaginal discs. The dpp and wg genes are expressed in the dorsal and ventral regions of the drosophila leg imaginal disc. A key factor in maintaining the non-overlapping regions of gene expression is the inhibition of dpp transcription by Wg signaling. The recent experimental studies of ventral leg disc cell showed the Arm/dTcf complexes activate the wg expression. In addition, the Arm/dTcf complexes cooperate with Brinker (Brk) to inhibit dpp transcription. Furthermore, ectopic expression of Arm or dTcf causes loss of dpp repression. To validate the experimental results and to determine the mechanism that leads to dpp repression, we developed an intra-cellular mathematical model to investigate the protein-protein and protein-DNA interactions. Our model showed that dpp repression occurred in response to Wg singaling only when Brk interacts with Arm to form a repressing complex R! 1TABR2 in the presence of non-repressing complexes. In the wing disc, Wingless signaling is affected by binding of the ligand to the glycosaminoglycan (GAG) chains of proteoglycans. GPI anchored heparan sulfate proteoglycans (HSPG) such as glypicans, antagonize wingless signaling by trapping it in the extracellular matrix. Syndecan (Sdc) is another kind of proteoglycan that is decorated with sugar chains. Current study shows that Drosophila Syndecan also modulates Wingless signaling but by a different mechanism. The one syndecan gene is transcribed to multiple isoforms with specific tissue distribution. Experimental results suggest that Syndecan antagonizes Wingless signaling by promoting the internalization of the secreted Wingless protein. Our extra-cellular mathematical model incorporates diffusion, reaction, internalization of ligand, ligand-receptor and ligand-nonreceptor to investigate such system. We found that when Sdc is overexpressed, loss of Wg signal occur! s and the amount of internal ligand complex increases.
We develop a deterministic mathematical model to describe reactivation of latent virus by chemical inducers. This model is applied to the reactivation of latent KSHV in BCBL-1 cell cultures with butyrate as the inducing agent. Parameters for the model are first estimated from known properties of the exponentially growing, uninduced cell cultures. The model is then extended to describe chemically induced KSHV reactivation in latently infected BCBL-1 cells. Additional parameters that are necessary to describe induction are determined from fits to experimental data from the literature. Our model provides good agreement with two independent sets of experimental data.
To examine the network-level encoding used by hippocampus to achieve its real-time representations of episodic information, we have used a 96-channel array to simultaneously record the activity patterns of as many as 260 individual neurons in the mouse hippocampus during various startling episodes. We find that the mnemonic startling episodes triggered firing changes in a set of CA1 neurons in both startle-type and environment-dependent manners. Pattern classification methods reveal that these firing changes form distinct ensemble representations in a low-dimensional encoding subspace. Application of a sliding window technique further enabled us to reliably capture not only the temporal dynamics of real-time network encoding but also postevent processing of newly formed ensemble traces. Our analyses revealed that the network-encoding power is derived from a set of functional coding units, termed neural cliques, in the CA1 network. The individual neurons within neural cliques exhibit ''collective cospiking'' dynamics that allow the neural clique to overcome the response variability of its members and to achieve real-time encoding robustness. Conversion of activation patterns of these coding unit assemblies into a set of real-time digital codes permits concise representation and categorization of discrete behavioral episodes.
We study the problem of closure of a hierarchy of coupled integro-differential equations describing the dynamics of the product densities m[k],k=1,2,..., arising from the deterministic approximation of a locally regulated, spatial birth-and-death point process, with rules motivated by plant population dynamics. In our approach, we truncate at order k=2 and close the product density m[3] through maximisation of the entropy functional of the process, restricted to a small (but not infinitesimal) observation window A, in such a way that the likelihood of observing configurations involving more than three points in A is negligible and A captures the scale of spatial correlations of third order. The maximization is carried out subject to the constraints of normalization and fixed product densities m[1] and m[2], that are given by the hierarchy. We thereby obtain an implicit, Kirkwood-type closure for m[3] , coupled to an ancillary integral equation for the domain of triplet correlations A. This new closure is enhanced with previously unnoticed correction terms, which vanish for the homogenous Poisson process. They are shown to become important as the process deviates from complete spatial randomness. Numerical solutions of the product density dynamics, based on this new closure, are in remarkably good agreement with the equilibrium values from individual-based simulations, both for aggregated and segregated spatial patterns.
Angiogenesis, the formation of new blood vessels from pre-existing
vessels, is a normal process in growth and development, as well
as in wound healing. Recent experimental work showed the importance
of the capillary sprouts in the formation of a new micro-vasculature
in a wound. Hence, we focus our attention on sprouting angiogenesis
and blood microcirculation in a healing wound.
We introduce an integrodifference equation model to study the spatial spread of epidemics through populations with overlapping and non-overlapping epidemiological generations. Our focus is on the existence of travelling wave solutions and their minimum asymptotic speed of propagation c*. We contrast the results here with similar work carried out in the context of ecological invasions. We illustrate the theoretical results numerically in the context of SI (susceptible-infected) and SIS (susceptible-infected-susceptible) epidemic models.
Even though much is known about the biophysics, anatomy and physiology of basal ganglia networks in Parkinson's disease, there is no clear understanding of how all these properties are connected to each other and what is the origin of the motor symptoms, including parkinsonian tremor. We present the development of biophysically based mathematical model of the basal ganglia-thalamocortical circuits of the brain, which are involved in the generation of pathological tremor. The model suggests the mechanism of how the cellular properties of pallidal and subthalamic neurons may give rise to tremor oscillations in the presence of thalamocortical feedback, thus providing insights into pathophysiology of parkinsonian tremor.
An averaged ODE model is presented to explore the effects of macrophage infiltration in tumours. The tumour is portrayed as an ensemble of three interacting populations: proliferating cells, quiescent cells and necrotic material. The process of vascularization is also taken into account and macrophages are chemotactically attracted into the tumour. Different scenarios are considered: either the quiescent cells or the necrotic material could promote the chemotactic recruitment and macrophages can be engineered to have therapeutic effects such as cytotoxicity or inhibition of angiogenesis. A bifurcation analysis is presented exposing qualitative differences in the dynamics of the possible submodels, e.g. the onset of oscillatory behaviour and the appearence of multiple stable stationary solutions. Special attention is payed to those contrasting behaviours which seem more relevant when devising future laboratory experiments.
FtsZ, a bacterial homolog of tubulin, forms single-stranded polymers
that assemble cooperatively. Models for cooperative polymerization
traditionally require polymers to be multistranded, suggesting that
new models are now needed. Potentially, cooperativity might emerge
if a subunit changes from a low to a high affinity conformation
when in contact with adjacent subunits in the polymer. Computer
programs that model chemical reactions could determine whether such
models for single-stranded polymers produce the lags and critical
concentrations indicative of cooperativity. However, conventional
programs are difficult to apply to polymers because an unlimited
number of different polymer species may exist, whereas these programs
require discretely defined species. Additionally, conventional programs
cannot efficiently track polymer lengths, the spatial relationships
between subunits within a filament or make rate constants dependent
upon them. We have developed a computer program that can track the
behavior of 10,000 subunits based on probabilities of reaction per
unit time. The modeling strategy uses a stochastic reaction event
generator. To validate the algorithm, we have successfully simulated
mathematically solvable polymerization schemes, including isodesmic
polymerization and the model for double-stranded polymerization
used to fit FtsZ data1. Our program is robust, eliminating feedback
loops that can occur in deterministic models when all long polymers
are combined into one chemical class. Additionally, our program
can track changing polymer length distributions over time. We are
currently extending the program to determine whether cooperativity
in single-stranded FtsZ polymers can be explained by models of polymerization
that include nucleotide hydrolysis, or in which a subunit's reaction
rates are tied to the chemical states of its neighbors.
Dendrites of many nerve cells are complex, branching structures that receive and process thousands of synaptic inputs from other neurons. Loci for receiving inputs are served by dendritic spines that are equipped with excitable channels. Here we develop a mathematical model of branched dendritic tree based upon a generalisation of the analytically tractable Spike-Diffuse-Spike model. The active membrane dynamics of spines are modelled by an integrate-and-fire process. The spines are assumed to be discretely distributed along a passive branched dendritic structure. The model supports saltatory travelling wave propagation and wave scattering amongst the dendritic branches. This model is ideally suited for the study of spatio-temporal filtering properties and neural response to different patterns of synaptic input.
An important problem in computational biology is the modeling of
several types of networks, ranging from gene regulatory networks
and metabolic networks to neural response networks. In [LS], Laubenbacher
and Stigler presented an algorithm that takes as input time series
of system measurements, including certain perturbation time series,
and provides as output a discrete dynamical system over a finite
field. Since functions over finite fields can always be represented
by polynomial functions, one can use tools from computational algebra
for this purpose. The key step in the algorithm is an interpolation
step, which leads to a model that fits the given data set exactly.
Due to the fact that biological data sets tend to contain noise,
the algorithm leads to over-fitting.
The tumor suppressor p53 is critical to ensure genomic stability. In single cells, the oscillatory p53 response to ionizing radiation (IR), which induces double stranded breaks (DSBs), is "digital," in that the number of oscillations rather than the amplitude shows dependence on the radiation dose. We present a model of single cell p53 dynamics in response to ionizing radiation. In our model, DSB sites interact with a limited pool of DNA repair proteins, forming DSB-protein complexes at DNA damage foci. Both the initial number of DSBs and the DNA repair process are modeled stochastically. The model assumes that the persisting complexes are sensed by ataxia telangiectasia mutated (ATM) kinase, which transduces in an ON/OFF manner the DNA damage signal to the downstream negative feedback oscillator consisting of p53 and its negative regulator Mdm2, a transcriptional target of p53. Our model exhibits coordinated oscillations of p53 and Mdm2 upon IR stimulation, with a stochastic number of oscillations whose mean increases with IR dosage, in agreement with the observed response of p53 to DNA-damage in single-cell experiments.
The study of the motility of cilia and flagella is of great importance
in biology and medicine. Mathematical modeling to simulate the internal
mechanism of cilia and flagella has attracted many researchers.
But no satisfactory work has been done for multiciliary beating.
We present an integrative computational model of multiciliary beating.
The axoneme is modeled by Dillon-Fauci approach which was first
used in [1] in 2000. Dillon-Fauci axoneme model is simple but very
successful in modeling the cilia and sperm motility (see [1,2]).
This model, based upon the immersed boundary method (Peskin), couples
the internal force generation of the molecular motors through the
passive elastic structure with the external fluid mechanics governed
by the Navier-Stokes equations. In our model, the multiciliary configuration
by the immersed boundary method does not cause much extra cost in
computation. In our numerical results, we will show how a single
cilium interacts with its n! eighboring cilia, how viscosity effects
its beating frequency, how metachronal wave, synchronization phenomena
are genereated. The challenging problems wherein will also be discussed.
At last we present the computer simulations for chlamydomonas swimming
and muco-ciliary interactions as applications of our model. This
project is a joint work with Professor Lisa Fauci, Tulane University
and Robert Dillon, Washington State University.
Streaming Video: Real Media Animals, including humans, use interaural time differences (ITDs) that arise from of different sound path lengths to the two ears, as a cue of horizontal sound source location. The nature of the neural code for ITD is still controversial. Current models advocate either a map-like place code of ITD along an array of neurons, consistent with a large body of data in the barn owl, or a rate-based population code, consistent with data from small mammals. Recently, it was proposed that these different codes reflect an optimal coding strategy that depends on head size and sound frequency. The chicken makes an excellent test case because its physical prerequisites are similar to small mammals, yet it shares a more recent common ancestry with the owl. We show here that, like in the barn owl, the brainstem nucleus laminaris in mature chickens displayed the major features of a place code of ITD. The physiological range of ITDs was systematically represented in the maximal responses of neurons along each isofrequency band. This is in contrast to the predictions from optimal coding theory and thus re-opens the question as to what determines the neural coding strategies for ITDs, including which code might be implemented by the human brain.
Streaming Video: Real Media I will describe some recent work in our group on the dynamics of the actin cytoskeleton in relation to the movement of a motile cell. First, I will describe work (joint with Adriana Dawes, Eric Cytrynbaum, and Bard Ermentrout) on a simple 1D spatial model of a cell. We show how the branching of actin filaments and the forces they exert on the cell membrane account for the protrusion velocity and characteristic actin density profiles. (This work is partly analytical and partly numerical.) We use this model to understand how branching rates and other biochemical parameters control cell speed by studying the relevant travelling wave solutions.I will also describe efforts (joint with AFM Maree, Alexandra Jilkine, Adriana Dawes and Veronica Grieneisen) at assembling a more detailed 2D spatial model of a crawling cell, in which we take into account the regulatory role of a set of signalling proteins (Cdc42, Rac, Rho). We show how the interplay between these and the actin cytoskeleton accounts for the ability of the cell to self-organize, polarize, maintain a stable shape and speed, and respond to new external signals.
What aspects of a stimulus cause a neuron to fire? How do stimuli affect the time of spikes? In this talk, I will discuss what we can learn about neuronal firing patterns by regarding neurons as nonlinear oscillators. The spike-triggered average or reverse correlation method is a common approach for determining what kinds of stimuli make a neuron fire. The poststimulus time histogram is another experimental measurement for describing the affect of a stimulus on the firing pattern of a neuron. The latter can be related to the former by using some optimality arguments. Both of these curves should be affected by the membrane properties of the individual neuron of interest. Since this is a huge-dimensional space, we will focus on one property of neurons which has been shown to be tightly coupled to neuronal dynamics: the phase resetting curve (PRC). The PRC describes the shift in the timing of a spike due to a brief stimulus as a function of the time since the last spike. We show that under certain circumstances there is a 1:1 mapping between the STA, the PSTH, and the PRC. Thus, we connect internal dynamics of neurons with their preferred stimuli and their population responses. This work is joint with Boris Gutkin, Alex Reyes, Nathan Urban, Roberto Galan, and Nicolas Fourcaud.
Streaming Video: Real Media This lecture will be devoted to a discussion of some of the basic problems in modeling actin dynamics. Actin polymerization and network formation are key processes in cell motility. Numerous actin binding proteins controlling the dynamic properties of actin networks have been studied and models such as the dendritic nucleation scheme have been proposed for the functional integration of at least a minimal set of such regulatory proteins. However, a complete understanding of actin network dynamics is still lacking. Even at the actin-filament level, the dynamics of the distribution of filament lengths and nucleotide profiles are still not fully understood. We will describe recent work on the evolution of the distribution of filament lengths and nucleotide profiles of actin filaments, both from a deterministic and a stochastic viewpoint. If time permits we will discuss work aimed at integrating microscopic models of actin dynamics into cell-level descriptions of motility.
Presentation materials: PPT Streaming Video: Real Media The main function of the circulatory system is to transport and exchange substances throughout the body. Delivery of oxygen is a particularly demanding function, because oxygen is relatively insoluble in water. Within blood vessels, oxygen is carried convectively by hemoglobin molecules within red blood cells. Oxygen exchange with tissue occurs by diffusion in the microcirculation, an extensive branching network of microscopic vessels that brings blood close to all oxygen-consuming tissues. The microcirculation regulates blood flow according to changing local demands over short and long time scales. Mathematical models can be used to gain insight into these processes. Models will be described for the mechanics of blood flow in capillaries, for oxygen exchange between blood and tissues and for structural adaptation of blood vessels. Applications to disease states including cancer will be discussed.
Presentation materials: PPT Streaming Video: Real Media The first generation of models for electrical activity in pancreatic beta-cells focused on ionic mechanisms. Negative feedback by calcium, directly onto calcium-activated potassium channels and indirectly onto ATP-sensitive potassium channels and sodium pumps, is the main type of mechanism considered in current models. Such models do a good job of accounting for the oscillations on a wide range of time scales, ranging from 10 seconds to about 2 minutes. However, even slower oscillations, with periods of 4 or even 10 minutes are often observed, and these often appear with the faster oscillations layered on top. This suggests that there is an additional mechanism for oscillations, which we have proposed is based on oscillations of glycolysis. We will discuss how two relatively simple oscillators, which can be off, oscillating, or tonically on, depending on stimulation level, can be combined to account for the great diversity of observed patterns. We will also consider the impact of metabolic oscillations on synchronization of beta-cells within the islet of Langerhans. Diffusion of glycolytic metabolites provides an important mechanism for secretion, but can also lead to oscillator death and a source of bistability.
In this presentation, I will demonstrate how quantitative modeling can not only shed light on the spatio-temporal growth of gliomas but also can have specific clinical application in real patients. Integration of our quantitative model with the T1-weighted and T2-weighted magnetic resonance (MR) imaging characteristics of gliomas can provide estimates of the extent of invasion of glioma cells peripheral to the imaging abnormality. Additionally, further insight can be gained from parametric mapping of kinetic model parameters derived from positron emission tomography (PET) with novel tracers. In summary, although current imaging techniques remain woefully inadequate in accurately resolving the true extent of gliomas, quantitative modeling provides a new approach for the dynamic assessment of real patients and helps direct the way to novel therapeutic approaches.
In cardiac ventricular myocytes, events crucial to excitation-contraction (EC) coupling take place in spatially restricted microdomains known as dyads. The length-scale over which this Ca2+ signaling occurs is a few tens of nanometers and the time-scale of these events spans the range of µsecs to msecs. Quantitative understanding of the functional consequences of these signaling events therefore requires development of models that are applicable over a range of spatio-temporal scales. We will present several new approaches for developing such multi-scale models of EC coupling. We will begin our analyses at the nano-scale level by presenting a model of dyad Ca2+ dynamics in which the Fokker-Planck equation is solved for the probability P(x, t) that a Ca2+ ion is located at position x at time t (Tanskanen et al, SIAM J. MMS, in press). The model will describe: a) dyad geometry; b) membrane surface charges; c) geometry and space-filling properties of the RyR (cryo-em), L-Type Ca2+ channel (LCC) and calmodulin proteins (crystal structures); d) stochastic gating of and Ca2+ flux through LCCs; and d) Ca2+ binding to RyR activation/inactivation sites, stochastic gating and Ca2+ flux through RyRs. Using this model, we will demonstrate that: a) Ca2+ signaling in the dyad is mediated by ~ tens of Ca2+ ions; b) these signaling events are noisy due to the small number of ions involved; and c) the geometry of the RyR protein may function to restrict the diffusion of and to "funnel" Ca2+ ions to Ca2+ activation | ||