F. Kloosterman, S. Layton, Z. Chen and M. Wilson
Bayesian decoding of unsorted spikes in the rat hippocampusJournal of NeurophysiologyVol. 111 No. 1 (2014) pp. 217-227 (Published)
An overview of Bayesian methods for neural spike train analysisComputational Intelligence and Neuroscience (special issue on â€œModeling and Analysis of Neural Spike Trainsâ€?) (In Press)
Z. Chen, S. Gomperts, J. Yamamoto and M. Wilson
Neural representation of spatial topology in the rodent hippocampusNeural Computation (In Press)
S. Adams, C. Zhang, H. Zambrano and T. Conlisk
Antibody-antigen binding in a flow-through microfluidic device51st AIAA Aerospace Sciences Conference (2013) (Published)
R. Cressman and V. Krivan
Two-patch population models with adaptive dispersal: the effects of varying dispersal speedsJ. Math. Biol.Vol. 67 (2013) pp. 329â€“358 (Published)
AbstractThe population-dispersal dynamics for predatorâ€“prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predatorâ€“prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.
N. Beckman and H. Rogers
Consequences of Seed Dispersal for Plant Recruitment in Tropical Forests: Interactions within the SeedscapeBiotropica (Under Revision)
AbstractSeed dispersal sets the stage for the suite of biotic and abiotic interactions that determine the fate of individual seeds. In this review, we first focus on how dispersal influences the â€˜seedscapeâ€™, or the combination of abiotic, biotic, and spatial factors that affect the probability of germination once a seed has reached its final location. We review recent papers that examine the effect of dispersers on (1) the quality of the habitat in which a seed lands; (2) the distance seeds are dispersed from the parent tree; and (3) the density and composition of plants within the neighborhood of a seed following deposition. Next, we explore methods used to scale these processes up to the level of populations. We highlight demographic models that integrate across multiple life history stages and predict the impact of dispersal in variable environments on population growth. We also review studies that analyze existing spatial patterns of trees within large forest plots and use various strategies to infer the processes that led to those patterns. We continue to scale up from populations to communities, and discuss three approaches that have been taken to understand how dispersal may affect diversity and abundance in the community. We finish by highlighting several areas of research that are particularly promising for future directions of study.
M. Schwemmer and T. Lewis
The robustness of phase-locking in neurons with dendro-dendritic electrical couplingJournal of Mathematical Biology (2012) (Published)
AbstractWe examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance ( gc ) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of gc are calculated from the experimentally measured coupling coefficient ( CC ) based on a single-compartment description of a neuron, and therefore may be severe underestimates of gc . With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of CC , which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.
K. Liao, X. Bai and A. Friedman
The role of CD200-CD200R in tumor immune evasionJ. Theor. Biol. (2013) (Accepted)
AbstractCD200 is a cell membrane protein that interacts with CD200 receptor (CD200R) of myeloid lineage cells. During tumor initiation and progression, CD200-positive tumor cells can interact with M1 and M2 macrophages through CD200-CD200R-compex, and downregulate IL-10 and IL-12 productions secreted primarily by M2 and M1 macrophages, respectively. In the tumor microenvironment, IL-10 inhibits the activation of cytotoxic T lymphocytes (CTL), while IL-12 enhances CTL activation. In this paper, we used a system approach to determine the combined effect of CD200-CD200R interaction on tumor proliferation by developing a mathematical model. We demonstrate that blocking CD200 on tumor cells may have opposite effects on tumor proliferation depending on the â€œaffinityâ€? of the macrophages to form the CD200-CD200R-complex with tumor cells. Our results help understanding the complexities of tumor microenvironment.
H. Kang, M. Crawford, M. Fabbri, G. Nuovo, M. Garofalo, P. Nana-Sinkam and A. Friedman
A mathematical model for microRNA in lung cancerPLoS OneVol. 8 No. 1 (2013) (Published)
AbstractLung cancer is the leading cause of cancer-related deaths worldwide. Lack of early detection and limited options for targeted therapies are both contributing factors to the dismal statistics observed in lung cancer. Thus, advances in both of these areas are likely to lead to improved outcomes. MicroRNAs (miRs or miRNAs) represent a class of non-coding RNAs that have the capacity for gene regulation and may serve as both diagnostic and prognostic biomarkers in lung cancer. Abnormal expression patterns for several miRNAs have been identified in lung cancers. Specifically, let-7 and miR-9 are deregulated in both lung cancers and other solid malignancies. In this paper, we construct a mathematical model that integrates let-7 and miR-9 expression into a signaling pathway to generate an in silico model for the process of epithelial mesenchymal transition (EMT). Simulations of the model demonstrate that EGFR and Ras mutations in non-small cell lung cancers (NSCLC), which lead to the process of EMT, result in miR-9 upregulation and let-7 suppression, and this process is somewhat robust against random input into miR-9 and more strongly robust against random input into let-7. We elected to validate our model in vitro by testing the effects of EGFR inhibition on downstream MYC, miR-9 and let-7a expression. Interestingly, in an EGFR mutated lung cancer cell line, treatment with an EGFR inhibitor (Gefitinib) resulted in a concentration specific reduction in c-MYC and miR-9 expression while not changing let-7a expression. Our mathematical model explains the signaling link among EGFR, MYC, and miR-9, but not let-7. However, very little is presently known about factors that regulate let-7. It is quite possible that when such regulating factors become known and integrated into our model, they will further support our mathematical model.
J. Hu, H. Kang and H. Othmer
Stochastic analysis of reaction-diffusion processesBulletin of Mathematical Biology (Accepted)
AbstractReaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially-discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.
H. Kang, L. Zheng and H. Othmer
A new method for choosing the computational cell in stochastic reaction-diffusion systemsJournal of Mathematical BiologyVol. 65 No. Series 6-7 (2012) pp. 1017-1099 (Published)
AbstractHow to choose the computational compartment or cell size for the stochastic simulation of a reactionâ€“diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reactionâ€“diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.
H. Kang and T. Kurtz
Separation of time-scales and model reduction for stochastic reaction networksAnnals of Applied ProbabilityVol. 23 No. 1 (2013) pp. 529-583 (Published)
AbstractA stochastic model for a chemical reaction network is embedded in a one-parameter family of models with species numbers and rate constants scaled by powers of the parameter. A systematic approach is developed for determining appropriate choices of the exponents that can be applied to large complex networks. When the scaling implies subnetworks have different time-scales, the subnetworks can be approximated separately, providing insight into the behavior of the full network through the analysis of these lower-dimensional approximations.
L. Hu, D. Chen and G. Wei
High-order fractional partial differential equation for molecular surface constructionMolecular based Mathematical Biology (2012) (Published)
AbstractFractional derivative or fractional calculus plays a signifcant role in theoretical modeling of scientiď¬?c and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
D. Chen, J. Roda, C. Marsh, T. Eubank and A. Friedman
Hypoxia inducible factors mediated-inhibition of cancer by GM-CSF: A mathematical modelBulletin of Mathematical Biology (2012) (Published)
AbstractUnder hypoxia, tumor cells, and tumor-associated macrophages produce VEGF (vascular endothelial growth factor), a signaling molecule that induces angiogenesis. The same macrophages, when treated with GM-CSF (granulocyte/macrophage colony-stimulating factor), produce sVEGFR-1 (soluble VEGF receptor-1), a soluble protein that binds with VEGF and inactivates its function. The production of VEGF by macrophages is regulated by HIF-1Î± (hypoxia inducible factor-1Î±), and the production of sVEGFR-1 is mediated by HIF-2Î±. Recent experiments measured the effect of inhibiting tumor growth by GM-CSF treatment in mice with HIF-1Î±-deď¬?cient or HIF-2Î±-deď¬?cient macrophages. In the present paper, we represent these experiments by a mathematical model based on a system of partial differential equations. We show that the model simulations agree with the above experiments. The model can then be used to suggest strategies for inhibiting tumor growth. For example, the model qualitatively predicts the extent to which GM-CSF treatment in combination with a small molecule inhibitor that stabilizes HIF-2Î± will reduce tumor volume and angiogenesis.
D. Chen and A. Friedman
A two-phase free boundary problem with discontinuous velocity: Application to tumor modelJournal of Mathematical Analysis and Applications (2012) (Published)
AbstractWe consider a two-phase free boundary problem consisting of a hyperbolic equation for w and a parabolic equation for u, where w and u represent, respectively, densities of cells and cytokines in a simpliď¬?ed tumor growth model. The tumor region Ω(t) is enclosed by the free boundary Γ(t), and the exterior of the tumor, D(t), consists of a healthy normal tissue. Due to cancer cells proliferation, the convective velocity ~v of cells is discontinuous across the free boundary; the motion of the free boundary Γ(t) is determined by ~v. We prove the existence and uniqueness of a solution to this system in the radially symmetric case for a small time interval 0 t T, and apply the analysis to the full tumor growth model.
A. Lam and Y. Lou
Evolution of Conditional Dispersal: Evolutionarily Stable Strategies in Spatial ModelsJournal of Mathematical Biology (2013)
AbstractWe consider a two-species competition model in which the species have the same population dynamics but dierent dispersal strategies. Both species disperse by a combination of random diusion and advection along environmental gradients, with the same random dispersal rates but dierent advection coecients. Regarding these advection coecients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we and that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be at least three evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.
Quikr: a Method for Rapid Reconstruction of Bacterial Communities via Compressive SensingOxford Journal of Bioinformatics (2013) (Under Review)
AbstractMany metagenomic studies compare hundreds to thousands of environmental and health-related samples by extracting
and sequencing their 16S rRNA amplicons and measuring their similarity using beta-diversity metrics. However, one of the first steps- to classify the operational taxonomic units withing the sample - can be a computationally time-consuming task since most methods rely on computing the taxonomic assignment of each individual read out of tens to hundreds of thousands of reads.
We introduce Quikr: a QUadratic, K-mer based, Iterative, Reconstruction method which computes a vector of taxonomic assignments and their proportions in the sample using an optimization technique motivated from the mathematical theory of compressive sensing. On both simulated and actual biological data, we demonstrate that Quikr is typically more accurate as well as typically orders of magnitude faster than the most commonly utilized taxonomic assignment technique (the Ribosomal Database ProjectĂ˘â?¬â?˘s Naive Bayesian Classifier). Furthermore, the technique is shown to be unaffected by the presence of chimeras thereby allowing for the circumvention of the time-intensive step of chimera filtering.The Quikr computational package (using MATLABor Octave) for the Linux and Mac platforms is available at http://sourceforge.net/projects/quikr/.
R. Azencott , A. Beri, Y. Gadhyan, N. Joseph, C. Lehalle and M. Rowley
Realtime market microstructure analysis: online Transaction Cost AnalysisQuantitative Finance (2013) (Submitted)
AbstractMotivated by the practical challenge in monitoring the performance of a large number of algorithmic trading orders, this paper provides a methodology that leads to automatic discovery of the causes that lie behind a poor trading performance. It also gives theoretical foundations to a generic framework for real-time trading analysis. Academic literature provides different ways to formalize these algorithms and show how optimal they can be from a mean-variance, a stochastic control, an impulse control or a statistical learning viewpoint. This paper is agnostic about the way the algorithm has been built and provides a theoretical formalism to identify in real-time the market conditions that influenced its efficiency or inefficiency. For a given set of characteristics describing the market context, selected by a practitioner, we first show how a set of additional derived explanatory factors, called anomaly detectors, can be created for each market order. We then will present an online methodology to quantify how this extended set of factors, at any given time, predicts which of the orders are under performing while calculating the predictive power of this explanatory factor set. Armed with this information, which we call influence analysis, we intend to empower the order monitoring user to take appropriate action on any affected orders by re-calibrating the trading algorithms working the order through new parameters, pausing their execution or taking over more direct trading control. Also we intend that use of this method in the post trade analysis of algorithms can be taken advantage of to automatically adjust their trading action.
R. Azencott , A. Beri, A. Jain and I. Timofeyev
Sub-sampling and Parametric Estimation for Multiscale DynamicsCommunications in Mathematical Sciences (2013) (To Appear)
AbstractWe study the problem of adequate data sub-sampling for consistent parametric estimation of unobservable stochastic differential equations (SDEs), when the data are generated by multiscale dynamic systems approximating these SDEs in some suitable sense. The challenge is that the approximation accuracy is scale dependent, and degrades at very small temporal scales. Therefore, maximum likelihood parametric estimation yields inconsistent results when the sub-sampling time-step is too small. We use data from three multiscale dynamic systems, the Additive triad, the Truncated Burgers-Hopf models, and the model with the Fast-Oscillating Potential to illustrate this sub-sampling problem. In addition, we also discuss an important practical question of constructing the bias-corrected estimators for a fixed but unknown value of the multiscale parameter.
Stochastic mechano-chemical kinetics of molecular motors: a multidisciplinary enterprise from a physicistâ€™s perspectivepp. 370 (Submitted)
AbstractAmolecularmotor ismade of either a singlemacromolecule or amacromolecular
complex. Just like their macroscopic counterparts, molecular
motors â€œtransduceâ€? input energy into mechanical work. All the nanomotors
considered here operate under isothermal conditions far from equilibrium.
Moreover, one of the possible mechanisms of energy transduction,
called Brownian ratchet, does not even have any macroscopic counterpart.
But, molecular motor is not synonymous with Brownian ratchet; a large
number of molecular motors execute a noisy power stroke, rather than
operating as Brownian ratchet. We review not only the structural design
and stochastic kinetics of individual single motors, but also their coordination,
cooperation and competition as well as the assembly of multimodule
motors in various intracellular kinetic processes. Although all
the motors considered here execute mechanical movements, efficiency and
power output are not necessarily good measures of performance of some
motors. Among the intracellular nano-motors, we consider the porters,
sliders and rowers, pistons and hooks, exporters, importers, packers and
movers as well as those that also synthesize, manipulate and degrade
â€œmacromolecules of lifeâ€?. We review mostly the quantitative models for
the kinetics of these motors. We also describe several of those motordriven
intracellular stochastic processes for which quantitative models are
yet to be developed. In part I, we discuss mainly the methodology and
the generic models of various important classes of molecular motors. In
part II, we review many specific examples emphasizing the unity of the
basic mechanisms as well as diversity of operations arising from the differences
in their detailed structure and kinetics. Multi-disciplinary research
is presented here from the perspective of physicists.
A. Sharma and D. Chowdhury
Error correction during DNA replicationPhysical Review EVol. 86 No. 011913 (2012) (Published)
AbstractDNA polymerase (DNAP) is a dual-purpose enzyme that plays two opposite roles in two different situations
during DNA replication. It plays its a normal role as a polymerase catalyzing the elongation of a new DNA
molecule by adding a monomer. However, it can switch to the role of an exonuclease and shorten the same
DNA by cleavage of the last incorporated monomer from the nascent DNA. Just as misincorporated nucleotides
can escape exonuclease causing a replication error, the correct nucleotide may get sacrificed unnecessarily by
erroneous cleavage. The interplay of polymerase and exonuclease activities of a DNAP is explored here by
developing a minimal stochastic kinetic model of DNA replication. Exact analytical expressions are derived for
a few key statistical distributions; these characterize the temporal patterns in the mechanical stepping and the
chemical (cleavage) reaction. The Michaelis-Menten-like analytical expression derived for the average rates of
these two processes not only demonstrate the effects of their coupling, but are also utilized to measure the extent
of replication error and erroneous cleavage.
A. Beri, D. Chowdhury and H. Jain
Agent-based and Macroscopic PDE models for foraging dynamics of competing ant colonies, and associated boundary interactions(2012) (In Preparation)
R. Azencott , A. Beri, Y. Gadhyan, N. Joseph, C. Lehalle and M. Rowley
Realtime Market Microstructure Analysis: Online Transaction Cost Analysis(2012) (In Preparation)
A. Beri, R. Azencott and I. Timofeyev
Calibration of Stochastic Volatility Model under Indirect Observability of the Volatility Process(2012) (In Preparation)
H. Kang, T. Kurtz and L. Popovic
Central limit theorems and diffusion approximations for multiscale Markov chain models(In Preparation)