My research is mainly in applied probability and analysis. See below for more detailed descriptions of some previous and current projects.
Dynamics of Molecular Motors
In cells, a major issue is the transport of cellular cargo from one area to another. Here “cargo” can be genetic material, protein, or really any large molecule. We’re working at the microscale, so it’s technically possible for cargo to move through diffusion due to thermal fluctuations. For especially long cells (some neurons are over a meter long) , back of the envelope calculations give estimates on the order of years for traveling across the cell by diffusion alone.
To deal with these time constraints, eukaryotic organisms have intricate configurations of microtubules, which could be thought of as a cell’s highway system. Molecular motors are taxi cargo along microtubules, with one end of the motor attached to the microtubule and the other to the cargo. Through a mechano-chemical process, two motor heads perform a stepping mechanism at the cost of ATP-ADP hydrolysis. Motors come in a variety of flavors, with various directions, speeds, and diffusivities. Often, several motors, of possibly different types, are attached to a single cargo. A single motor during procession may completely attach or detach from the microtubule. The complex behavior that can arise from different motor types and their possible states is of interest to both biologists and mathematicians.
A cartoon of two molecular motors transporting a cargo along a microtubule. (Picture taken from “Asymptotic analysis of microtubule-based transport by multiple identical molecular motors”- Journal of Theoretical Biology (2012))
My most recent work, with Peter Kramer (RPI) and John Fricks (ASU), estimates effective behavior from multiple motors attached to a single cargo. Effective behavior can mean things such as expected run time before detachment, as well as average velocity and diffusivity assuming arbitrarily long run lengths. For our model, we coarse grain stepping behavior of the microtubule heads, and model motor and cargo behavior along the microtubule as a system of stochastic differential equations (SDEs).
As mentioned before, motors can be in either attached or detached states. For each possible detachment state, we have a different set of nonlinear SDEs to work with. To complicate matters further, the transition rates between detachment states depend on relative positions of cargo and motors, and are modeled by Cox processes, or Poisson processes with random rate functions. Our goal is to find explicit expressions of long term statistics for such a system, but a this point things can get quite complicated!
To approximate the complicated behavior describe with a simpler system, we use multiscale averaging. We can find “fast” and “slow variables through nondimensionalization, by a simulation by Euler-Maruyama discretization is also illuminating. Here’s a figure which shows behavior for two kinesin motors attached to a cargo at an attachment and detachment event:
For each figure, we map time against position of motors or cargo along a microtubule. Top two figures: positions before and after a transition from two to one attached motors (detachment event). Bottom figures: positions before and after transition from one to two motors (attachment event).
Comparing the relative frequency of oscillations between variables, unattached motors are the fastest, attached motors are the slowest, and cargo oscillations lie somewhere in between. This separation of scales gives us license to perform averaging arguments, which estimate behavior of fast variables as constant coefficient SDEs. We can perform similar arguments with attachment and detachment rates to produce a Markov chain approximation to motor behavior. At this point, we appeal to renewal theory arguments to obtain law of large numbers and central limit theorem type approximations for effective behavior of ensemble positions.
In two dimensions, gas diffusion in foams and the heating of polycrystalline metals can be modeled as a trivalent network which evolves under mean curvature. The following picture shows the process at three different times:
Three snapshots of the coarsening process
This system is interesting in multiple ways. First, the gathering of statistics for grain topologies (the number of sides of each grain) is highly nontrivial. During the coarsening process, grains with less than six sides annihilate due to the Von-Neumann “n-6 rule”, which states that grains with n sides have a rate of area growth proportional to n-6. After vanishing, neighboring grains change topologies to keep the network trivalent.
Left side: grains before annihilation. Right side: grains following annihilation. Note the top row of pictures, where topologies can also change due to a side collapsing.
Thus, a natural clock for this system, describing the rate of topology change, is the number of deleted grains.
After a short time, however, something unexpected occurs: statistics of side topologies stabilize!
Distribution of topology: the dotted line is the stationary distribution obtained from direct curvature flow. The solid lines show evolution of the kinetic model with an imposed diffusive term, corresponding to increased side switching.
Currently, there is no rigorous proof of universality for this grain statistic. A solution from curvature flow currently seems intractable. In fact, there is still debate about how we should reassign grain topologies following the disappearance of a single grain. One common method, which converts the problem of curvature driven parabolic equations to transport equations, makes a “gas approximation”, or a mean field model where the notion of neighbors is eliminated. Most of these models are written in their final hydrodynamic limit, without addressing whether finite models approach the limiting equation, or even if the limiting equations themselves are well-posed.
My research with this problem starts with a finite dimensional stochastic process, simulating the evolution of grains. Particles drift on multiple copies of the positive real line. Each copy represents a topology, and a particle’s position on the line represents area. Here’s the main picture:
In this instance, a grain with three sides is annihilated. Three grains are randomly chosen to lose a side (denoted by vertical arrows).
This model allows for significant generalizations, with respect to both the number of lines, or “species”, and the set of rules for random reassignment after a “boundary event” occurs, where a particle a annihilated. The natural model for this is the “piecewise deterministic Markov process” (PDMP). Here’s a PDMP occurring on four species:
In this instance of a PDMP, a particle annihilates on species 2. By predetermined rules, three randomly chosen particles are assigned to new species.
Click here for a preprint of the main paper studying the generalized “k-species model”, along with this study for a fully worked example on a minimal model on one species, which gives laws of large numbers for total loss of particles, as well as a general Glivenko-Cantelli type theorem for two-phase sampling.
My first project at Brown was on the optimality of self-assembling nets. In collaboration with scientists from the Gracias Lab at Johns Hopkins University, we defined a gluing algorithm to create a configuration space of folding states. Viewed as a Markov chain, we derived a set of birth death equations which predict prevalent pathways that minimize defects.
Left: The path space for an octahedron and its non-convex cousin: the “boat”. Right: a single net with multiple end states.
How can you improve performance in a neural network? A simple answer is to add more edges to its architecture, the underlying topology which connects hidden layers. In this preprint, we test the accuracy of CrossNets, a neural network which allows for connections in a general directed acyclic graph, against other networks, and formally prove convergence of the network to a local minimum. This is work with Chirag Agarwal, a graduate student in electrical engineering.
Left: A minimal neural network with a lateral connection z within a hidden layer. Right, networks with such connections can be represented by more layers. The resulting architecture is always an instance of a directed acyclic graph.