Description:

    To a first approximation the mechanics of many biological systems can be modeled by elastic structures which interact with a fluid.  The immersed boundary method (21) has been demonstrated to be an efficient numerical method for this purpose and has been used in studying a variety of biological phenomena including: blood flow in the heart (20), lift generation in insect flight (16), and wave propagation in the cochlea of the inner ear (11).  With advances in experimental methods in cell and molecular biology there is a strong  interest in modeling both qualitatively and quantitatively systems at microscopic length scales (1).  On the length scale of cells and cell organelles thermal fluctuations become significant and play an important role in many cellular processes (1,19,24). 

    In the paper (3, 13), an extended  immersed boundary method has been formulated which incorporates thermal fluctuations in terms of stochastic partial differential equations which have appropriate stochastic forcing terms consistent with statistical mechanics.   The stochastic immersed boundary method (3) also overcomes stiffness in the equations allowing for long time steps which under-resolve the stochastic dynamics of the fluid on small time scales while accounting for important statistical contributions over a time step, see the papers (3,6).  Here a few computational projects related to the stochastic immersed boundary method are discussed.  For a more mathematical and complete discussion see the papers (3 -- 7).

    In the stochastic immersed boundary method (3) only the degrees of freedom associated with the fluid are stochastically forced.  The immersed structures fluctuate by virtue of their coupling to the fluid.  While the Gaussian random field (10) used to force the fluid equations were derived to obtain Botlzmann statistics for the fluid (23,14), this derivation was done  in the absence of immersed structures (3).  An important consideration for the method is whether the immersed structures (for conservative forces) also have fluctuations obeying Boltzmann statistics.  As a basic test, a single particle confined within a spherical chamber with a soft-wall potential was simulated (3).  Below is a plot of a sample trajectory of the simulation and the soft-wall potentials used.  The exact Boltzmann distribution can be computed analytically as a function of the radial coordinate and is given in the last row of the figure below.

    For this basic model, it was found that the fluctuations appear to have Boltzmann statistics, see the figure below.

     A natural approach to analytically study the equilibrium statistics of the method is to consider the Forward-Kolomogorov Equations (10,18) associated with the stochastic process.  The equilibrium distribution typically corresponds to the steady-state solution of an elliptic equation.  For the stochastic immersed boundary method a number of technical issues arise as a consequence of only the fluid degrees of freedom being stochastically forced.  In fact, the steady-state Forward-Kolomogorov Equations for the stochastic immersed boundary method are  degenerate and not elliptic because the immersed structure degrees of freedom only have derivative terms of first order from the pure advection.  However, while the equations fail to be elliptic they can be shown to fall into the broad class of "hypoelliptic" equations in the sense of Hormander (9,17).  Along with Jon Mattingly of Duke University, I am studying the invariant measures of the stochastic immersed boundary method using results for hypoelliptic equations. 

     As a further check on the method, simulations of diffusing particles have been performed to check that the correct scaling is obtained in the physical parameters.  This can also be shown analytically, and that in particular an Einstein relation holds for particles simulated with the stochastic immersed boundary method provided an appropriate effective friction is associated with the particles, see paper (5).  

    An interesting feature of the stochastic immersed boundary method is that the hydrodynamics are simulated explicitly.  On small time scales the fluctuations of a Brownian particle deviate from Einstein's classical theory (2).  Since the particle is immersed in a viscous fluid the fluctuation of a particle in a given direction imparts momentum on the surrounding fluid which then influences later motion of the particle for a short period of time.  This "hydrodynamic memory" first observed in molecular dynamics simulations in (2) manifests itself in the velocity correlation function of a Brownian particle as a slow algebraic decay of order t^-3/2, as opposed to an exponential decay as in the classical theory.  It has been shown both analytically and through simulation in (7) that the stochastic immersed boundary method captures this effect for Brownian particles, see the figure below (blue,  numerical simulations, green, predicted t^-3/2 power law), the deviation for long times occurs because of a finite size effect, see paper (7).

     In the stochastic immersed boundary method, approaches from Ito's calculus (10,18) have been used to analytically handle the stochastic dynamics of the fluid (3).  This allows for long time steps with under-resolve the dynamics of the fluid while accounting for statistical contributions over the time step.  An important question is the accuracy of the numerical method for both time steps which resolve the dynamics of the fluid on small time scales and for time steps which under-resolve the fluid dynamics.  In the paper (6), numerical analysis has been carried out for the method for time steps which can be broadly classified as small, intermediate, or large relative to the smallest time scales of the fluid.   The asymptotic expressions obtained in this analysis (6) have been validated by numerical simulations in the absence of a structure force, see the figures below (left, small time steps, right large time steps).

     These simulations show that the numerical method has first order accuracy for immersed structures when small time steps are used which fully resolve the fluid dynamics (3).  For long time steps, a different notion of accuracy must be used since the time step is never taken to zero, see paper (3).  For the time steps which under-resolve the fluid dynamics, the method exhibits a scaling proportional to the time step size, but with an error which is small relative to the displacement of a structure in the exact solution over a time step (3).  In the figures is a comparison of the numerical error estimated from high resolution simulations (data points with error bars) with the analytically derived asymptotic expressions for the numerical error (dashed curve) .

    Below are a few examples showing how the stochastic immersed boundary method (3) may be applied in the context of a few basic models.  More sophisticated biological applications are also being carried-out in related work, see the project page.

Example 1:

     An interesting feature of the continuum formulation of the stochastic immersed boundary method is that the immersed structures are advected by a continuous flow by virtue of averaging the fluid velocity.  For continuous structures, such as spatial curves, the solution map for the configuration of the structure at time t is a homeomorphism.  Thus, topological features of the structure and the complement space are preserved by the flow.  For example, curves which form knots have their crossings preserved by the flow.  When discretizing in time, numerical errors are introduced with respect to this feature of the method.  An interesting question concerns the degree to which topological structures are preserved in simulation in the presence of these numerical errors.  Below are some examples of polymer knots and links simulated with the stochastic immersed boundary method.  These polymers retain their knotted structure for a long duration of time, even when control particles from distant segments of the curve become fairly close, but as a result of the numerical error eventually lose their topological structure.  Click on the images below to see movies taken form some of the numerical simulations.

Example 2:

     Knotted structures are thought to play an important role in biological systems, such as the knotted DNA confined with the nucleus of the cell or in virus capsids (26).   As a simple example, we shall show how the stochastic immersed boundary method can be used to compute the wall force (pressure) associated with confining a polymer knot to a spherical volume.  In this example, the excluded volume interactions are explicitly added to the model through a repulsive interaction between different segments of the polymer to take into account the thickness of the polymer and to avoid uncrossing of the knot.  Click on the images below to see movies taken form some of the numerical simulations.

     The wall pressure is computed and shows a decreasing trend, as one might expect intuitively on account of the reduction of the entropic penalty for confining the polymer as the number of knot crossings increase, see figure below (23).

Example 3:

     More complicated systems can also be simulated with the stochastic immersed boundary method.   Below is a simple example of how a Brownian Ratchet (19,24) towing a cargo vesicle can be simulated by the method.  In the model, there is a particle (red) which is restricted to a one dimensional track (blue / yellow), but free to diffuse otherwise.  The particle (red) when diffusing from the blue to the yellow region to the right is prevented from back-slipping to the blue region by a reflecting boundary condition.  The particle can then diffuse from the yellow to the blue region to the right with a similar reflecting boundary condition which prevents back-slipping and so on.  This stochastic process is referred to as a Brownian Ratchet.  One natural question for this system is the mean velocity at which the particle moves to the right.  This can be shown to depend on the diffusivity of the particle and the length of the interval (19).  When a cargo is attached to the Brownian Ratchet there are many interesting questions concerning how the cargo effects the stochastic dynamics of the particle and the mean velocity of the ratchet.  The Brownian Ratchet has been proposed as one theoretical mechanism that may be used by motor proteins and polymerization processes within the cell (19, 24).

    Other biological applications of the stochastic immersed boundary method include modeling the dynamics of biological membranes and osmotic effects, which is being carried-out in related work.  For a more mathematical and complete discussion of this work, see the papers (3 -- 7).

    Trefoil Knot (no excluded volume) [AVI]

    Two Rings (no excluded volume) [AVI]

    Unknot (excluded volume) [AVI]

    Trefoil (excluded volume) [AVI]

    Figure-eight knot (excluded volume) [AVI]

    Solomon knot (excluded volume) [AVI]

    Brownian ratchet towing a vesicle [AVI]

    Brownian ratchet transporting a polymer [AVI]