Fall 2007 Workshop for Young Researchers in Mathematical Biology (WYRMB): Abstracts and Lecture Materials
Plenary Speakers
In this presentation, I will highlight some of the challenges that have been brought up by emergent and re-emergent diseases. I will provide examples from Tuberculosis, Influenza and other infectious diseases.
The nervous system produces electrical activity at a variety of frequencies at all times, in a manner dependent on cognitive state and task. Two frequency bands associated with attention and learning are the gamma band (30-90 Hz) and the theta band (4-12 Hz), which appear together in the hippocampus during some kinds of learning. This talk discusses an in vitro network that displays both rhythms, with each frequency band produced by a subset of the network; the subsets overlap in their components and compete for control of the common portions. Dynamical systems modeling is used to explain how changes in the angle of the slice can change the power of the two frequencies, and how the cells producing the slower rhythm can coordinate the cells producing the faster one.
In this talk I will present recent research on the spatial dynamics of emerging wildlife diseases. I will analyse models for West Nile virus, both in a dynamical context and in a spatial context, using travelling wave theory to describe spread into new environments. I will then describe recent interdisciplinary work on the dynamics of naturally occurring parasites on wild salmon, namely sea lice, and the role played by salmon farms in changing those dynamics for juvenile salmon. In this case, the spatial patterns of infection can be used to deduce the source of disease and, ultimately, to assess the impact of aquaculture on the wild salmon population.
A fundamental issue in neurobiology is to understand how the central nervous system (CNS) can perform accurate and reliable calculations with neurons that are intrinsically variable and unreliable devices. This question has a long history in the auditory system where auditory nerve (AN) fibers of mammals typically show standard deviations of latency (time to firing) of about 1 msec under repeated trials with the same sound. Yet, psychophysical experiments have shown that mammals can detect extremely small binaural timing differences down to the low millesecond and even nanosecond range. This has been confirmed by physiological experiments that show that certain classes of neurons in the brainstem have very small standard deviations of latency under repeated trials, even though these neurons are receiving information only from these same auditory nerve fibers that are such sloppy timers. Work with Colleen Mitchell will be described that investigates the relationship between convergence of fibers and the sharpening of timing. This work uses a simple time-window model; the target neuron fires the first time it gets m hits in \ep milleseconds. But do real neurons have sharp time-windows?
Working at a national laboratory has much in common with working in academia but also has some differences. Federal employment carries with it certain legal obligations that can strange to an academic because we are always under scrutiny from our benefactors, the taxpayers. There are, however, many compensations, specifically at NIH, such as freedom from writing grants and teaching. The research environment is also different, with more of a mission orientation. Also, theoreticians need to adjust to being in an environment in which they are in a small minority, but they also get prime access to a vast array of experimental work that is conducted intramurally. In addition to my own group, other theory groups exist and are growing. The future is hopeful for young theoreticians who are open to these possibilities.
Life Sciences has been undergoing a shift from model rich and data poor to data rich and model poor. In addition, with increasing power to measure the dynamics of components of living system, the challenge of integrative understanding paradoxically grows. Furthermore, the major problems in Life Sciences cannot be solved through application of technology alone, as is the case of most of society's problems. Transdisciplinarity is based on development of a shared conceptual framework from which different disciplinary, social and other views can draw on to increase understanding and solve problems. Mathematics is the closest humans have come to esperanto. As such it plays a crucial role in transdisciplinary research. The need for advancement of theory and increasingly interwoven experimentation, modeling and simulation in Life Sciences continues to grow. Model-driven experimentation is one of the outcomes that are only possible if integration of modeling-simulation-theory-experiment are considered as the actual process of scientific inquiry. Where possible, this talk will try to exemplify how a transdisciplinary approach is affecting Life Sciences research and how central mathematics is to the development of common conceptual frameworks.
In difference to Turing-type or diffusion driven instabilities, as they are occur in the context of pattern formation related to diffusive morphogenes, mathematical models for structure formation due to local cell interaction are presented. In comparison, these can give rise to different structures. Mathematical and biological implications will be discussed in detail.
Membrane proteins are highly prevalent molecules which form over 30% of a currently known gene products. Despite their high abundance, very few membrane protein structures have been reported. Studying these large protein/membrane complexes using standard structural biology techniques, such as X-ray crystallography and solution state NMR, is often precluded. Solid state NMR offers a viable alternative.
In this talk, we will examine how oriented solid state NMR methods can be used to derive structural parameters. This process involves the interpretation of spectra and structure refinement, both of which make use of mathematical tools. These will be presented along with the assumptions and examples will be given.
Work done in collaboration with W.R.P. Scott.
MBI Talks
A discrete-time SIS patch model is presented. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. In low-risk habitats, the disease-free equilibrium (DFE) is stable when the mobility of infected individuals lies above a threshold value, but for high-risk habitats, the DFE is always unstable. When the DFE is unstable, there exists a unique endemic equilibrium (EE). When the EE exists, it tends to a spatially inhomogeneous DFE as the mobility of susceptible individuals tends to zero. Sufficient conditions for whether high-risk patches in the limiting DFE have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. This work is in collaboration with Linda J.S. Allen (Texas Tech) and Yuan Lou (Ohio State).
Poster Presentations
The stochastic simulation algorithm, or Gillespie Algorithm, is a tool used to simulate discrete biochemical reaction systems when there are a small to moderate number of molecules. The Gillespie Algorithm has been adapted to handle systems with delays (such as with gene transcription and translation) and approximations (such as tau leaping) have been developed in order to increase computation speed. In this poster I will show how viewing discrete biochemical systems as jump Markov processes modeled by independent unit Poisson processes leads to efficient and accurate modifications of the Gillespie Algorithm.
The dynamics of many coupled nephrons is being investigated in our studies. Based on nearest neighbor coupling, nephron models are coupled and studied both analytically and numerically. The analytical results show that, when the nephrons in the coupled system are identical, nephrons' flow rates are more likely to be in the oscillatory state. Furthermore, the evidence of phase synchronization and frequency entrainment has been observed among the nephrons' flow rates. The numerical results show that, when the nephrons in the coupled system are slightly different from each other, an almost similar type of qualitative dynamics exists between the total flow of the whole system and the total flow of a sub system governed by a sub-collection of nephrons. This strongly suggests that the nephrons with different flow oscillations may constantly interact and adjust each others' flow rates until the coupled system locks itself into a desired state.
The possible role of angiogenesis as a target for anti-cancer therapy was discovered by Folkman in 1972. However, its implementation is directly connected with anti-angiogenic drugs, which was discovered in nineties of the XX century by O'Reilly et al.
It is known that growth of tumors of diameter greater than 1--2 mm is supported by newly formed vessels, which is necessary to provide nutrients and growth factors and clear products of cellular metabolism. In the lack of nutrients, tumor cells secrete growth factors, one of the best known is vascular endothelial growth factor, which can stimulate endothelial cells to proliferate, migrate and form new blood vessels. However, the newly formed vessels have highly unstable structure and its stabilization crucially depends on maturation (coverage by pericytes) which is governed by another growth factor --- platelet-derived growth factor and the system of angiopoietins.
The considered process is very complicated. Mathematical modeling can bring a better understanding and lead to better treatment protocols (see Arakelyan et. al., Angiogenesis 5, 203--214, 2002). The first mathematical model of angiogenesis was proposed by Hahnfeldt et al. It is described as a system of ODE based on the Gompertz growth of tumor with carrying capacity equal to endothelial cells volume. It occurs that independently on the parameters there exists exactly one globally stable positive equilibrium. Therefore it cannot reflect possible instability of the blood vessel formation and structure that is observed in the experiments.
The aim of the work is to combine the ideas presented in paper by Agur et. al. ( Disrete Cont. Dyn. Sys. B 4, 29--38, 2004.) and Hahnfeldt and propose the three variable model with the tumor carrying capacity depending on vessel density. Due to the modifications our model can exhibit the stable behavior in the case without delay. On the other hand, there can exist more than one positive critical point in our model. After introducing time delay to the model Hopf bifurcation occurs and the oscillatory behavior is observed.
The motion of primary nodal cilia present in embryonic development resembles that of a precessing rod. This movement is also studied experimentally as a benchmark in conjunction with pulmonary cilia research. Implementing regularized Stokeslets to model fluid flow numerically simulates a situation for which colleagues have exact mathematical solutions and experimentalists have corresponding laboratory studies. Stokeslets are fundamental solutions to the steady Stokes equations, which act as external point forces when placed in a fluid. By strategically distributing regularized Stokeslets in a fluid domain to mimic an immersed boundary (e.g. cilium), one can compute the velocity and trajectory of the fluid at any point of interest. The simulation can be adapted to a variety of situations including passive tracers, rigid bodies, and numerous rod structures in a fluid flow generated by a rod either rotating around its center or its tip near a plane. Quantitative! and qualitative comparisons to theory and experiment have shown that a numerical simulation of this nature can generate insight into fluid systems that are too complicated to fully understand via experiment or exact numerical solution independently.
The vertebrate body plan is composed of the repetition of segments visible at the level of the vertebral column (e.g., vertebrae). This organization is first established during early embryogenesis through the periodic formation of embryonic segments called somites (120 min in mouse, 90 min in chick and 30 min in zebrafish). This process is associated with a molecular oscillator termed the segmentation clock which drives the periodic expression of so called cyclic genes in the tissue precursor of the somites, the presomitic mesoderm (PSM). Thus far, only a handful of cyclic genes, mostly linked to Notch signaling, has been identified. The goal of this work was to apply a genome-wide approach to identify systematically the cyclic genes and to provide insights about the regulatory network of this clock. We generated a microarray time series dataset of mouse PSMs during one clock oscillation cycle. We performed a mathematical (Lomb-Scargle) analysis to identify periodic expression profiles. As a result, we identified a large number of new cyclic genes linked to several signaling pathways and participating in negative feedback loops. When ordering the periodic profiles by phase, the cyclic genes linked to Notch and FGF signaling appeared to oscillate in antiphase with the ones related to Wnt signaling, suggesting a tight temporal coordination of these oscillating pathways. Finally, the candidate cyclic genes were validated experimentally and their inactivation often correlates with segmentation defects. As a conclusion, this quantitative and systematic analysis of the segmentation clock system by microarrays identifies a complex oscillating network. Based on microarray data recently generated, we are now comparing the clock regulatory networks across species ranging from fish to chick and mouse. These datasets are particularly suitable to test different mathematical methods of pattern detection as well as of network analysis.
Many emerging and reemerging diseases, such as rabies, SARS, Ebola, etc., use wild mammalian populations as reservoirs. Most of the outbreaks in humans are apparently a result of the spillover from those species. Therefore, the analysis of the health risk, associated with those diseases, is not possible without a good understanding of the disease dynamics in wild populations. The traditional compartmental epizootic models are relatively easy to implement and analyze, but usually impose unrealistic aggregating assumptions and incorporate parameters, which are not measurable in wild conditions. We propose a novel computational and analytical approach, which address the maintenance of the disease into mammalian populations through individual (within host) models of the response of the host to a virus challenge. The parameters in those models can be estimated, based on experimental infections in laboratory conditions. The dynamics of the individual models are used to d! efine realistic survival, susceptibility and transmission rate functions as well as to develop and parameterize epizootic models of the evolution of the disease into a uniform population, and disease demographics models, which are structured by the individual response governed by specific host system, such as the immune system. Applications of the proposed approach in modeling bat rabies are presented. Different environmental and physiological factors and their influence on the disease dynamics are also investigated.
Temperate zone bats are subject to serious energetics constraints due to their high surface area to volume relations, the cost of temperature regulation, the high metabolic cost of flight, and the seasonality of their resources. To my knowledge, there are no individual-based mathematical models for any bat species. First, an individual model is developed for a female bat primarily based on life history and energetics. This model describes the growth of an individual female bat using a system of differential equations modeling the dynamics of two main compartments: storage (lipids) and structure (proteins and carbohydrates). The model is used to test hypotheses related to strategies used by temperate bats to meet their energy demands. The model is parameterized for the little brown bat, Myotis lucifugus, because of information available on energy budgets and changes in body mass throughout its life history. However, with appropriate modifications the conceptualizat! ion might be applied to other species of bats with similar life histories. Second, the individual model is integrated into a structured population model. Characteristics of the individuals determine the structure and, subsequently the dynamics of the population. This methodology uses and integrates the information on bat biology and physiology that has been primarily collected at the individual level. Survival and reproductive rates estimated from simulated populations under varying density dependece are comparable to those reported in the literature for natural populations of M. lucifugus. The population model provides insight into possible regulatory mechanisms of bat populations sizes and dynamics of survival and extinction. A better understanding of population dynamics can assist in the development of management techniques and conservation strategies, and to investigate stress effects.
Spatially selective observations often arise in ecology where a random walking animal strays beyond the spatial region in which the experimental set-up allows detection, e.g., the line of sight of a camera, a radio transmitter, a microscope or a telescope. In ecological studies the linear dimensions of the window may range from a few kilometers, as in the study of tree diversity in a forest, to a few hundred meters when observing animals via trapping methods, to a few centimeters in the case of the motion of microorganisms in sea water. In the context of the estimation of animal home range we provide a theory that accounts for the limitation of the sampled area in that data gathered contain cutoffs beyond which no information can be collected. The theoretical tool used is the Fokker-Planck equation, its characteristic quantities being the diffusion constant which describes the motion of the animals, and the attractive potential which addresses their tendency to live near their burrows. The measurement technique is shown to correspond to the calculation of a certain kind of mean square displacement of the animals relevant to the specific probing window in space corresponding to the sampled region. The output of the theory is a sigmoid curve of the observable mean square displacement as a function of the ratio of distances characteristic of the home range and the observation region, along with an explicit prescription to extract the home range form observations. Applications of the theory to mark-recapture experiments of rodents in Panama and New Mexico are shown. Extension of the theory to the cases of animals that perform anomalous random walks is also given.
In this work we want to examine the effect of the residual stress on the stress distribution through the deformed arterial wall and on the material properties and geometry of the artery. It is known that there is a stress stored in an artery even if there is no external force, that is, a blood vessel ring opens up to a sector of some angle when it was cut radially. This residual stress has been taken into account in the recent developments of the arterial walls models. In contrast to those models where the stress-free configuration was the reference one, we assume that the load-free (intermediate) configuration is the reference configuration. The motivation for this is that the stress-free radial cut is fictitious. The material properties and geometry parameters of the arterial wall are obtained by solving an inverse problem which accounts for an eigenvalue problem for a self-adjojnt operator. The measured quantity is the pressure on the inner wall of the artery corresponding to frequencies of an ultrasonic transducer.
We develop a stochastic model that quantifies the dispersal of the slug parasitic nematode Phasmarhabditis hermaphrodita as a heterogeneous population. The model relies on a statistical analysis of features of the nematode trail determined from direct observations of individual Ts behaviour in homogeneous and heterogeneous environment. In both cases we notice a large variation in individual behaviour among the population reflected in nematode speed that is not constant within the population. In particular, we find that the nematode speed in heterogeneous structure follows a generalized gamma distribution. As a results, at population level, in both environments the nematode trail is characterized by strong temporal autocorrelation between steps in terms of step length. Based on these observations we build a long-memory stochastic process in 2d modelling diffusion in heterogeneous population, i.e. unlike the normal diffusion, here the diffusion coefficient is not co! nstant among the population but follows a gamma distribution. Computed explicitly in terms of modified Bessel functions of second kind, the probability density function associated to this process shows that nematode dispersal follows a leptokurtic distribution.
We analyze a biophysical model of an entorhinal cortex layer V pyramidal cell that includes persistent sodium and slow potassium as non-standard currents using reduction of dimension and dynamical systems techniques to determine the mechanisms for the generation of mixed-mode oscillations. We have found that the standard spiking currents (sodium and potassium) play a critical role in the analysis of the interspike interval. To study the mixed-mode oscillations, the six dimensional model has been reduced to a three dimensional model for the subthreshold regime. Additional transformations and a truncation have led to a simplified model system with three timescales that retains many properties of the original equations, and we employ this system to elucidate the underlying structure and explain a novel mechanism for the generation of mixed-mode oscillations based on the canard phenomenon.
In particular, we show the existence of a special solution, a primary canard, that serves as a transition between mixed-mode oscillations and spiking in the singular limit. Additionally, we conjecture that the canard solution persists for sufficiently small values of the singular perturbation parameter and provide numerical evidence for the conjecture.
A biological process involves highly complex network of metabolic pathways, most of which are unknown and uncovered. Biologists and mathematicians work separately and together to solve the puzzle. In the recent years, advances in experimentation and technology have opened doors for following the dynamics of a system in real-time. This data is also called the time course data of a dynamically changing metabolic pathway. Information about the interactions of various metabolites is hidden in this data. This information can be difficult to extract using conventional analytical techniques. Each metabolic pathway is operated by several groups of enzymatic mechanisms which in turn are composed of elementary chemical reactions that obey mass action kinetics. An approach that assembles the metabolic pathway from the elementary chemical reactions by intelligent processing of selecting the right reactions can provide an answer to solving this puzzle. Global non-linear modeling technique can make this approach possible.
We have developed a new method based on global-nonlinear modeling to infer reaction mechanism from time course data. Our method involves two steps: (a) proposition of a family of model elementary chemical reactions and (b) parsimonious model selection and fitting of the data. In the later step, a synergistic process that controls the model size and the best fit forms the intriguing aspect of the method. The technique can be modified and implemented to several types of time series data namely, simple chemical kinetics, complex metabolic pathway and recently the genetic micro-array. The poster will illustrate the new method we have developed to infer reaction mechanisms from time series data obtained from experiments and its future perspectives.
We study the coexistence of multiple periodic solutions for a quadratic integrate-and-fire neuron model (QIF) of recurrent inhibitory loops, which incorporates two important biological features --- the firing procedure and absolute refractoriness. We show that the interaction of the delay, the feedback and the refractoriness can generate three basic types of oscillations and these three basic oscillations can then be pinned together to form interesting coexisting periodic patterns. We derive general principles that determine whether a periodic pattern can and should occur and we apply such principles to some detailed case studies. In particular, we show how pattern transitions occur at certain critical time delays and how these transitions yield the coexistence of multiple pattern subsets in certain subintervals.
Proposals have been made to use a transposable element (TE) to drive a disease resistance gene into a vector population. We investigate the feasibility of these proposals using a continuous-time multi-type branching process model. The model assumes: (a) in the early stages of spread the number of hosts carrying the TE is small and hence all matings involve at least one host not carrying the element; (b) replicative transposition rate decreases dependent on the number of TEs in each host; and (c) the fitness of each host decreases with the number of TEs it contains. These assumptions are used to determine the conditions under which an introduced element has a significant probability of spreading through an Anopholes gambiae host population at a rate acceptable to public health goals. The model favors a transgenic release immediately following the dry season when the host population begins to grow. Increasing the number of transgenic hosts released has the greatest influence on reducing the probability that the TE will be lost from the population. Following release, the rate at which the element increases its proportion in the host population is most sensitive to its replicative transposition rate. The model recommends a replicative transposition rate greater than 0.1 per element per generation to satisfy public health goals. The model suggests that empirical research should initially be directed at accurately estimating the replicative transposition rate when there is only a single TE in the genome.
Biofilms form when bacteria adhere to surfaces in moist environments by excreting a slimy glue-like substance. The human body is heavily colonised by microbes and to a large extent this isn't a problem but sometimes the presence of bacteria creates significant difficulties. Swarming (the rapid spread of bacterial colonies on a moist semi- solid substrate) plays an important part in many bacterial infections, including lung infections in, for example, cystic fibrosis patients. We aim to develop an understanding of the processes involved in bacterial swarming with the eventual aim of using this knowledge to compare different treatment strategies. Our work involves developing a model of swarming bacteria constrained within a thin film. The equations describing the biological mechanisms determining the behaviour of the bacteria are coupled with the standard thin-film reduction of the Navier-Stokes equation (which describes the action of the biosurfactant on the fluid boundary). We present some results from this modelling, along with a comparison of these results with the available experimental data.
Osteochondral defects are holes in articular cartilage that result from degradation of the tissue extracellular matrix due to osteoarthritis. Biocompatible hydrogel scaffolds have potential utility for filling such defects to restore mechanical integrity and facilitate cell proliferation and tissue regeneration in the defect region. One design goal was to maximize the mechanical properties of the hydrogel at physiological temperature. Potential Elastin-like Polypeptide hydrogels were screened using Artificial Neural Networks to determine an optimal hydrogel. Models of cartilage regeneration in a hydrogel scaffold seeded with cartilage cells (chondrocytes) are also formulated and analyzed for dynamical interactions between nutrients, hydrogel and extracellular matrix constituents in an evolving gel-tissue construct.
In response to hypoxia, the transcription factor hypoxia-inducible factor 1 (HIF1) activates over 100 genes involved in mammalian cell homeostasis. HIF1 has been called a "master switch" of angiogenesis, and its regulation has been linked to cancer metastases, insensitivity to radio- and chemotherapy, inflammation, and the severity of injury after ischemic stress. While experiments have driven knowledge of HIF1 since its discovery in 1992, there are many remaining questions inaccessible in vivo or in vitro. To test hypotheses relating to HIF1 regulation and quantify downstream effects, we are developing computational models representing the HIF1 system on multiple biological levels. Current models include kinetic molecular level models, a network signaling representation, and a phenotypic, cellular level model of angiogenic sprouting. The molecular model consists of kinetic equations mapping the molecular steps in HIF1a protein degradation in normoxia, HIF1a synthesis in chronic hypoxia, and effects of the enzyme and cofactors involved in the HIF1a hydroxylation pathway. The model is applied to cellular redox states associated with normal conditions, chronic hypoxia, and ischemia. The network model, consisting of biological circuit representations, allows the molecular model to be used to represent changes in the HIF1 pathway occurring in cells during hypoxia and normoxia, driven by kinases as well as oxygen levels. Finally, a cellular rule-based model, relates hypoxia to blood vessel sprouting, through the HIF1-dependent expression of the growth factor VEGF, hypoxia-dependent cell proliferation, cell migration, and the VEGF-dependent expression of Notch ligand Delta-like 4. These models were used to address hypotheses as to what are the major determinants of HIF1 regulation at the transcriptional and molecular level, and how this regulation may affect phenotypic response at the cell and vessel level. Computational modeling of the HIF1 system offers new detailed, quantitative insight into the complex mechanisms underlying hypoxic-response at the microvasculature.
We study the population-level behavior of a swarm of boids comprised of naive and informed individuals, for which a sound macroscopic description has remained elusive. This model can be regarded as a measure-valued generalization of the classical velocity jump process. Interestingly, the ensamble behavior of the swarm resembles closely that of the Advection Diffusion Equation with Memory. This memory can be fit parametrically by a Mittag-Leffler function. We describe a procedure to obtain the macroscopic dynamics of an ensamble of swarms via a generalized master equation, that leads to the ADEM.
The only known predictor of cardiac ventricular arrhythmia is a complex pattern in the electrocardiogram (ECG) waveform, which may be linked to instabilities in the dynamic release and reuptake cycle of calcium (Ca) in the ventricular cardiomyocyte. Understanding this system would provide possible targets for future drug development or gene therapy in the treatment of cardiac arrhythmia. Although existing computer simulation models can reproduce many details of Ca dynamics such as graded release and alternans of release, such models are highly parameterized and can be diffucult to interpret. Here we constuct an alternative low-dimensional mathematical model that captures the essential dynamics of Ca cycling as it is released out of the sarcoplasmic reticulum (SR) via clusters of ryanodine receptor (RyR) channels. Particular attention is paid to local stochastic properties of both the initial triggering current (via L-type Ca channels) and the RyR cluster response in computing the probability of Ca spark activation. We use a novel computational model of the cell emphasizing the nature of small-size receptor clusters, the spatial distribution of microdomain release sites, and their neighbor-to-neighbor cooperativity to formulate a theoretical/analytical understanding of the breakdown of normal calcium release-and-uptake cycle during repeated pacing stimulation. Although our resulting equations describe Ca dynamics at a reduced whole-cell level, their derivation incorporates the effects of small-size stochasticity, unlike traditional mean-field equations.
We consider the buffered bistable system as a model for the fertilization Ca{2+} wave, which is a simplified version of the Li and Rinzel model (Journal of Theoretical Biology 166:461-473 (1994)) by Smith, Pearson, and Keizer (Computational Cell Biology Springer-Verlag, 101-139 (2002)) with an crucial improvement: the kinetics of the buffers are not necessarily fast with respect to the Ca{2+} reactions. Our main concern is to show that buffers retain many important qualitative features of the unbuffered model (e.g., the wave activity, the stability properties of the associated steady states, and the threshold phenomenon). This also answers the following open problem partially: Although adding significant quantities of exogeneous calcium buffers (fluorescent dyes) to the system would change the quantitative properties of the system (the speed and shape of waves, for instance), do they change the qualitativie properties of the system?
We use mathematical modeling to quantitatively characterize a novel short-lived cyan fluorescent protein. We show that this characterization allows us to infer the underlying transcriptional response from fluorescent measurements, thereby providing a tool for monitoring transcript levels in single cells. Next we use stochastic modeling to investigate how the fluorescence maturation time and protein half life influence fluctuations in fluorescence levels. Our analysis reveals that for proteins with short-half lives fluorescence measurements can over-estimate fluctuations in protein levels, whereas for long-lived reporters fluorescence measurements typically underestimate these fluctuations. Our investigations also illustrate that fluctuations in fluorescence levels depend in a nontrivial way on the maturation time of fluorescence protein.
Protein structures determined by NMR (Nuclear Magnetic Resonance Spectroscopy) are not as fixed and detailed as those by X-ray crystallography, due to insufficient distance data available from NMR experiments. For instance, an NMR determined protein is generally represented by an ensemble of energy-minimized structures with low resolutions. Therefore, the applications of NMR determined structures are severely limited in some important fields, such as homology modeling and rational drug design. We have proposed a knowledge-based approach to refine NMR-determined structures. The method has been developed based on analyzing the statistics of pairwise inter-atomic distances and deriving distance-based potential mean force (PMF). It shows that the structures are improved significantly in terms of several standard criteria when a selected set of distances are optimized with corresponding potential mean force.
On the other hand, the multiple-structural alignment of structures of an NMR ensemble is critical to study the protein function, motion and domains in the solution. However, the superimposition could be distorted if the optimal alignment is obtained purely from mathematical perspective. We introduce a novel computational method, using weighted RMSD calculations. In this method, the protein fluctuations are determined by Gaussian Network Model and considered as weight factors. Then the optimal superimposition will be not geometric-based but dynamic-based, using weighted RMSD calculations. For NMR ensembles, the iterative weighted RMSD algorithm has been developed. Our results show that the superimposition of NMR ensembles can be greatly improved in terms of correlations with protein fluctuations and hence can provide a better understanding of NMR determined protein structures in the solution.
Motile bacteria, like human beings, follow the rule of the nature, responding to environmental cues and moving toward more favorable locations. The movement of bacteria in response to changes of specific metabolites and signaling molecules, as well as pH, temperature, light, salinity, and oxygen state in their surroundings, is called bacterial chemotaxis. How can they sense information, amplify/transfer signals, initiate responses, and finally adapt to environmental changes? In a best studied bacterium Escherichia coli (E. coli), these questions have been basically answered from the view of molecular biology. However, many mysteries still remain to us, for example, origin of high sensitivity. Currently, the sensitivity is widely believed to come from 1) phosphorylation-dependent interaction of chemotactic protein CheY with clustered chemoreceptor/CheA/CheW complexes upon chemotactic stimulation, and 2) allosteric interaction of phosphorylated CheY with flagellar protein FliM at the base of flagellar motors.
Our project used an mathematical modeling and computer simulation based approach to study the E. coli chemotaxis. First, we developed a basic model without considering chemoreceptor complex clustering, which can accurately reproduce the excitation and adaptation profiles of chemotaxis. Then we focused on quantitative analysis of the role of chemoreceptor complex clustering in E. coli chemotaxis to explain high sensitivity in the signal transduction pathway. Based on the recent findings of our collaborator Dr. John S. Parkinson at University of Utah, assuming that chemoreceptor clusters are comprised of signaling teams, in the unit of trimer of receptor dimers, which contain different receptor types and act collaboratively, we developed a static ~trimer of dimer T model, analyzing the contribution of the structural and functional unit ~trimer of dimers T to signaling sensitivity. The prediction of the fraction of active chemoreceptors as a function of ligand concentration from the ~trimer of dimers T model is in good agreement with the experimental findings. Our work provides a clear ~quantitative T picture of the signaling pathway in E. coli chemotaxis, theoretically indicates that clustering of chemoreceptor complexes could be the primary source of high sensitivity, and is helpful in understanding human infectious disease caused by E. coli.
Proteus mirabilis is an enteric bacterium. It is well known for its ability to swarm over hard surfaces forming concentric ring patterns, and for causing severe urinary tract infections. Recent experimental results show that swimmer cells stream inward toward the inoculation site, and form a number of complex patterns, including radial and spiral streams, rings and traveling trains. To understand the underlying mechanism of these patterns, we developed a hybrid cell-based model, which incorporates a simplified single cell signal transduction model. By assuming that swimmer cells respond to a chemoattractant that they produce, we predict the formation of radial streams as a result of the modulation of the local attractant concentration by the cells. We further predict the spiral streams by incorporating a swimming bias of the cells near the surface of the medium.
Transcription is the first step in gene expression, and the step at which most gene regulation occurs. Transcription consists of 3 distinct stages: initiation, elongation, and termination. Out of all these steps, elongation is the process most amenable to a quantitative description; experimental results in the past 5 years have made it possible to test predictions from quantitative models.
In this poster, a chemical kinetic model of the transcriptional elongation dynamics of RNA polymerase along a DNA sequence is introduced. The proposed model governs the discrete movement of the RNA polymerase along a DNA template, with no consideration given to elastic effects. The model's novel concept is a "look-ahead" feature, in which nucleotides bind reversibly to the DNA prior to being incorporated covalently into the nascent RNA chain. Analytical AND computational results for the proposed model are presented for specific DNA sequences used in actual single-molecule experiments of RNA polymerase along DNA. By replicating the data analysis algorithm from the experimental procedure, the computational model produces velocity histograms, enabling direct comparison with these published experimental results. Finally, parameter estimation results of the model to experimental data, along with their interpretation, are discussed.
Synaptically coupled neurons show in-phase or anti-phase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty in the coexistence of multiple stable oscillatory solutions. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the former, where a rich variety of co-existing solutions including localized oscillations occur. We construct these solutions via a multi-scale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions co-exist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution that coexists with other in-phase or anti-phase solutions is clearly represented in the stochastic phase dynamics.
We introduce a novel concept, the minimal molecular surface (MMS), for the theoretical modeling of biomolecule-solvent interfaces. When a less polar macromolecule is immersed in a polar environment, the surface free energy minimization occurs naturally to stabilize the system, and leads to an MMS separating the macromolecule from the solvent. For a given set of atomic constraints (as obstacles), the MMS is defined as one whose mean curvature vanishes away from the obstacles. Based on the theory of differential geometry, an iterative procedure is proposed to compute the MMS via the mean curvature minimization of molecular hypersurface functions. Extensive numerical experiments, including those with internal and open cavities, are carried out to demonstrate the proposed concept and algorithms. Comparison is given to the molecular surface. The proposed MMS is typically free of geometric singularities.
