overarching research interest is applying stochastic modeling and
analysis to characterize the behavior of complex systems, particularly
linking phenomena at a formally macroscopic or collective level to the
dynamics prescribed at an individual or formally microscopic
Stochastic models in microbiology
Molecular Motor Models
The fundamental actors in transporting organelles within cells are families of protein molecules, predominantly dynein and kinesin, which act as molecular-level engines, converting chemical fuel (ATP) into some sort of useful work, such as directed transport. Mathematical models for these molecular motors are typically stochastic, as thermal fluctuations play a strong role in influencing their dynamics, both by the noisy buffeting by the solvent molecules as well as the random diffusive transport of ATP to the proper binding location to activate the next chemomechanical cycle for progress. A great deal of mathematical and physical study of the mechanisms and performance of natural and synthetic molecular motors has taken place over the last 20 years. With Juan Latorre (Freie Universitaet Berlin) and Grigorios Pavliotis (Imperial College), we contributed to this body of work by applying homogenization theory and various techniques from stochastic analysis to map out how the transport effectiveness of a fundamental ``flashing ratchet" model depends on the governing parameters, and to evaluate the regimes in which simplified Markov chain models adequately represented the effective transport of the more detailed model. Over the last five years, a rapidly growing amount of theoretical and experimental work has been focusing on how the dynamics of a cargo is governed by multiple molecular motors bound to it simultaneously, as is expected to be relevant in vivo. Within a working group at the Statistics and Applied Mathematical Sciences Institute (SAMSI) we critically examined this literature, and with John Fricks (Penn State, statistics), Scott McKinley (Univ. Florida) and Avanti Athreya (Johns Hopkins), we formulated a new modeling framework for molecular motors cooperatively bound to a cargo which both accounted for stochastic spatial fluctuations in a more physically consistent manner than most of the existing models and allowed for analytical techniques to connect single-motor properties to those of the motor-cargo complex. We are now, together with experimentalist William Hancock (Penn State, bioengineering), working on developing our basic model to incorporate more realistic features, with the broad aim of linking detailed dynamical understanding of molecular motors at an individual level to larger scale experimental observations. During discussions of this effort at the Zentrum fuer Interdisziplinaere Forschung in Bielefeld, Germany, some of the other research fellows in residence, namely Leonid Koralov (Maryland), Yuri Makhnovskii (Topchiev Institute), and Leonid Bogachev (Leeds) have begun collaborating with me on developing some of the more technically challenging mathematical aspects needed to capture the relevant physics.
Synchrony in stochastically driven neuronal networksA grand challenge in theoretical neuroscience is to explain how higher-level brain functions, such as memory and object recognition, can be explained in terms of the known biophysics of the constituent neurons and their connections. The overwhelming complexity of the brain naturally frustrates a direct mathematical attack on this problem, so mathematical neuroscience has in recent years focused on more tractable analogue questions, whose resolution can bring insight into these broader conceptual issues. I became involved in mathematical neuroscience through co-advising former graduate student Katherine Newhall (now a Courant Instructor at NYU) with colleague Gregor Kovacic and collaborator David Cai (Shanghai Jiao-Tong), where we were concerned with a project of characterizing the presence of strongly synchronous behavior in a neuronal network model in terms of the individual neuron dynamics and the network model describing their connections. We employed the basic stochastically driven integrate-and-fire model for the neurons, and both simple and, with Maxim Shkarayev (William and Mary) complex scale-free network model architectures. Randomness enters into these models both to describe generic unstructured input signals (typically via canonical Poisson point processes) and to generate complex networks, for which only certain statistics are prescribed. Katherine Newhall, Peter Mucha (UNC), Amanda Traud (NC State), and I are exploring generalizations of these ideas to more general dynamical network models with cascades. During a recent visit to the Mathematical Biosciences Institute (MBI), Janet Best (Ohio State) and I formed a research group, together with two MBI postdocs and two of her graduate students, to pursue a related question of how the architecture of a metastable neuronal network model for a sleep-wake system affects the distribution of times spent in the sleep or wake states.
Suspensions of Swimming MicroorganismsOne of the most vigorous fields of recent inquiry regarding the connection of collective behavior to the rules of individuals is in the quantitative modeling of animal swarming or flocking. In fact for terrestial-scale organisms, for which much of this work is focused, the individual mobility and interaction rules are not yet precisely understood, and various hypotheses are under active exploration. Swimming microorganisms, on the other hand, mainly because of their rather limited cognitive powers, have comparatively very well understood mechanisms for motion as well as physical and biological rules for interaction. A growing range of experiments on suspensions of swimming microorganisms moreover reveals a range of remarkable collective behavior, such as patterned flows and changes to effective fluid properties such as viscosity. From a mathematical point of view, therefore, because of the sound theoretical foundation of individual dynamics and interactions and the range of well-controlled experiments on populations, swimming microorganisms serve as an excellent model system for developing analytical techniques for connecting collective dynamics to the individual dynamical rules. Moreover, beyond this theoretical motivation, the understanding of suspensions of swimming microorganisms has strong potential for applications to driving microfluidic systems. Together with Patrick Underhill (Chemical Engineering) and graduate student Kajetan Sikorski, we have been working on incorporating both statistical correlations and stochastic effects into a theory for predicting the effective behavior of a swimming microorganisms in a suspension, given the well-defined physical equations for individual motion and hydrodynamic interaction. The stochastic aspect here arises predominantly not from thermal fluctuations, but from coarse-graining of the detailed dynamics of the swimming process. An important part of this project is clarifying, on quantitative physical grounds, the source and impact of these stochastic components.
Stochastic network modeling in epidemiologyKristin Bennett and I are working with graduate student Lei Yao to develop a mathematical model and statistical technique for inferring unobserved transmission events from data of treated tubercolosis patients in New York State. The objective is to apply individual-based stochastic dynamical models for the spread of tubercolosis with computational methods from network tomography and phylogenetics to integrate temporal information from the data into Bennett's clustering methods.
Statistical Aspects of Multiscale ComputingDuring a visit to the Statistical and Applied Mathematical Sciences Institute (SAMSI) for their program on ``Stochastic Dynamics," a cross-disciplinary project emerged between computational scientist Sorin Mitran (University of North Carolina), myself, and statisticians M. Susie Bayarri
(Universitat de Valencia), James Berger (Duke), Murali Haran (Penn State, ), and Hans Rudolf Kuensch (ETH Zurich) from a contemporaneous program. We are exploring, with undergraduate student Sunli Tang, how to strengthen, using systematic techniques from the statistics community, the representation and interaction of the microscale and mesoscale elements of a novel multiscale simulation method (time-parallel continuum-kinetic-molecular) recently introduced by Mitran.
Simulation of Weakly Nonlinear Wave Turbulence
With former graduate student Warren Towne, Yuri L'vov (Rensselaer) and I have been developing an approach to simulating weakly nonlinear statistical systems which mitigates artifacts of periodic boundary conditions commonly employed.
Stochastic modeling in ecology
Lister (biology) and I have co-advised a number of undergraduate
students on theoretical ecology modeling projects as part of the
``Computational Science Training in the Mathematical Sciences''
program, and out of this experience, we have been developing
mathematical models for ecosystems which incorporate more biophysical
realism. In particular, we are working on an size-structured
model for daphnia with explicit coupling to nutrient concentrations,
applying a novel probabilistic model for computing the transfer of
individuals between size compartments. We are also developing
quantitative frameworks for how foraging decisions are affected by
partial information accumulated over time concerning predators.
Multiscale Random Field Simulation
Nonstandard Multiple Scale
Multiple scale asymptotics run through much of my research, usually in the context of the derivation of simplified effective equations to describe multiscale stochastic systems. I have been working with Christof Schuette (Freie Universitaet Berlin) and Jessika Walter (Ecole Polytechnique Federale de Lausanne) on such asymptotic mode reduction projects for biomolecular modeling, accounting particularly for metastable behavior. In this application, as well as in some of my other projects, the multiple scale asymptotics naturally involve three time scales. Together with Adnan Khan (Lahore University of Management Sciences) and Robert E. Lee DeVille (University of Illinois), I am working on clarifying some subtle aspects that arise in these kinds of asymptotic calculations in ordinary differential equation models.
J. C. Latorre, P. R. Kramer, and G. A. Pavliotis, "Numerical Methods for Computing Effective Transport Properties of Flashing Brownian Motors," submitted to Journal of Computational Physics (2013). (preprint version)
O. Kurbanmuradov, K. Sabelfeld, and P. R. Kramer, "Randomized Spectral and Fourier-Wavelet Methods for Multidimensional Gaussian Random Vector Fields," accepted to Journal of Computational Physics (2012). (preprint version)
K. A. Newhall, E. P. Atkins, P. R. Kramer, G. Kovacic, and I. R. Gabitov, "Random Polarization Dynamics in a Resonant Optical Medium," Optics Letters 38 (6), (2013): 893-895. (preprint version)
J. C. Latorre, G. A. Pavliotis, and P. R. Kramer, "Corrections to Einstein's relation for Brownian motion in a tilted periodic potential," Journal of Statistical Physics 150 (4), (2013): 776-803. (preprint version)E. P. Atkins, P. R. Kramer, G. Kovacic, and I. R. Gabitov, "Stochastic Pulse Switching in a Degenerate Resonant Optical Medium," Physical Review A 85 (2012), 043834. (preprint version)
S. A. McKinley, A. Athreya, J. Fricks, and P. R. Kramer, "Asymptotic Analysis of Microtubule-Based Transport by Multiple Identical Molecular Motors," Journal of Theoretical Biology 305 (2012): 54-69. (preprint version)
A. Porporato, P. R. Kramer, M. Cassiani, E. Daly, and J. Mattingly, "Local Kinetic Interpretation of Entropy Production through Reversed Diﬀusion," Physical Review E 84 (2011): 041142. (preprint version)
R. Keating, K. S. Smith, and P. R. Kramer, "Diagnosing lateral
mixing in the upper ocean with virtual tracers: Spatial and temporal
K. A. Newhall, G. Kovacic, P. R. Kramer, and D. Cai, "Cascade-Induced Synchrony in Stochastically-Driven Neuronal Networks," Physical Review E 82, (2010): 049103. (preprint version)
S. R. Keating, P. R. Kramer, and K. S. Smith, "Homogenization and Mixing Measures for a Replenishing Passive Scalar Field,"Phys. Fluids. 22, (2010): 075105. (PDF) Copyright 2010 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
K. A. Newhall, G. Kovacic, P. R. Kramer, D. Zhou, A. Rangan, and D. Cai, "Dynamics of Current-Based, Poisson Driven, Integrate-and-Fire Neuronal Networks," Communications in Mathematical Sciences 8 (2), 2010: 541-600. (preprint version)
P. R. Kramer, J. C. Latorre, and A. A. Khan, "Two Coarse-Graining Studies of Stochastic Models in Molecular Biology," Communications in Mathematical Sciences 8 (2), 2010: 481-517. (preprint version)
P. R. Kramer and S. R. Keating, "Homogenization Theory for a Replenishing Passive Scalar Field," Chin. Ann. Math. 30B (5), 2009: 631-644. (preprint version)
P. J. Atzberger and P. R. Kramer, "Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations," Mathematics and Computers in Simulation, 79, 2008: 379-408 (preprint version)
P. R. Kramer, C. S. Peskin, and P. J. Atzberger, "On the foundations of the stochastic immersed boundary method," Computer Methods in Applied Mechanics and Engineering, 197(25-28), 2008: 2232-2249 (preprint version)
P. R. Kramer, A. Khan, P. Stathos, and R. E. L. DeVille, "Method of Multiple Scales with Three Time Scales," Proceedings in Applied Mathematics and Mechanics 7(1), 2007: 2040031 - 2040032. (preprint version)
J. C. Latorre, P. R. Kramer, and G. A. Pavliotis, "Effective Transport Properties for Flashing Ratchets Using Homogenization Theory," Proceedings in Applied Mathematics and Mechanics 7(1), 2007: 1080501-1080502. (preprint version)
P. J. Atzberger, P. R. Kramer, and C. S. Peskin, "Stochastic immersed boundary method incorporating thermal fluctuations," Proceedings in Applied Mathematics and Mechanics 7(1), 2007: 1121401-1121402. (preprint version)
B. Baydil, P. R. Kramer, and K. S. Smith, "Parameterization for Mesoscale Ocean Transport through Random Flow Models," Proceedings in Applied Mathematics and Mechanics 7(1), 2007: 2150021 - 2150022.. (preprint version)
P. J. Atzberger and P. R. Kramer, "Theoretical Framework for Microscopic Osmotic Phenomena," Phys. Rev. E 75, 2007, 061125. (preprint version)
P. R. Kramer, O. Kurbanmuradov, and K. Sabelfeld, "Comparative
Analysis of Multiscale Gaussian Random Field Simulation
Algorithms," Journal of
Computational Physics226 (1), 2007: 897--924.
P. J. Atzberger, P. R. Kramer, and C. S. Peskin, "A stochastic immersed boundary method for
fluid-structure dynamics at microscopic length scales," Journal of Computational Physics
224 (2), 2007: 1255-1292. (preprint version)
E. Castronovo and P. R. Kramer, "Subdiffusion and Superdiffusion in Lagrangian Stochastic Models of Oceanic Transport," Monte Carlo Methods and Applications, 10 (3-4), 2004: 245-256. (preprint version)
L. J. Borucki, T. Witelski, C. Please, P. R. Kramer, and D. Schwendeman, "A theory of pad conditioning for chemical-mechanical polishing," Journal of Engineering Mathematics, 50 (1), 2004: 1--24. (preprint version)
P. R. Kramer and C. S. Peskin, "Incorporating Thermal Fluctuations into the Immersed Boundary Method," Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, K. J. Bathe, ed., Elsevier, 2, 1755--1758. (preprint version)
P. R. Kramer, J. A. Biello, and Y. Lvov, ``Application of weak turbulence theory to FPU model,'' Discrete and Continuous Dynamical Systems, Expanded Volume for the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, 2003: 482--491. (preprint version)
J. A. Biello, P. R. Kramer, and Yuri Lvov, ``Stages of energy transfer in the FPU model,'' Discrete and Continuous Dynamical Systems, Expanded Volume for the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, 2003: 113--122. (preprint version)
P. R. Kramer and A. J. Majda, "Stochastic mode reduction for immersed boundary method," SIAM J. Appl. Math. 64 (2), 2003: 369--400. (PDF)
P. R. Kramer and A. J. Majda, "Stochastic mode reduction for particle-based simulation methods for complex microfluid systems," SIAM J. Appl. Math. 64 (2), 2003: 401--422. (PDF)
P. R. Kramer, A. J. Majda, and E. Vanden-Eijnden, "Closure approximations for passive scalar turbulence: A comparative study on an exactly solvable model with complex features," J. Stat. Phys., 111 (3/4), 2003: 565--679. (PDF)
P. R. Kramer, "Two different rapid decorrelation in time limits for turbulent diffusion," J. Stat. Phys., 110 (1/2), 2003: 87--136. (PDF)
A. J. Majda and P. R. Kramer, "Simplified models for turbulent diffusion: Theory, numerical modelling and physical phenomena," Physics Reports, 314 (4-5), 1999: 237-574. (PS)
P. R. Kramer, "A review of some Monte Carlo simulation methods for turbulent systems,'' Monte Carlo Methods and Applications, 7 (3--4) 2001: 229--244. (PDF)