We performed simulations considering the dynamic progress of the junction between the virion and the cell surface to be a stochastic, Markov process 
. The system itself consists of a rigid sphere (the 100nm-diameter virion) interacting with a deformable surface (the plasma membrane, 200×200nm in dimension). The virion is populated with gp120 trimers, which can bind receptors (CD4 and CCR5) located on the plasma membrane (). All entities within the system including the plasma membrane, the configurations of viral proteins, the position of the virion itself, the dynamics of the bonds between the virus and the cell, as well as the positions of the cellular receptors (which are either bound or unbound to gp120), are specified by discrete states in Markov dynamics. The probabilities of generating a particular sequence of states, or a trajectory, are governed by the transition rates between these states. The transition rates between states that involve changes in the physical configuration of the system (i.e. the movement of the virion, the positions of the proteins and the plasma membrane from their current position to each possible new position) were calculated according to the local-steady state approximation of the Fokker-Planck equation 
is the forward association rate for the
possible state adjacent to the current state (
is the diffusion coefficient of each physical entity of the system (membrane proteins, viral proteins, etc.) whose position has changed between the current and future state,
is the change in position of that physical parameter,
is the difference in system energy which results from the movement of that physical parameter between states and
is Boltzmann constant and
is the absolute temperature. This approximation calculates the rate of transition between the current state and subsequent adjacent state using the total change in energy that accompanies the progression from one discrete state to another. Specifically, a favorable change in energy (e.g. the relaxation of a bond between viral proteins and cellular receptors) results in an increased transition rate and an unfavorable change in energy (e.g. the deformation of the plasma membrane) results in a decreased transition rate from one state to the next.
Each possible state differs from the current state by the discrete position of any physically real object within the system (e.g. the x, y, z position of the virion, the position of a CD4 protein or a CCR5 protein, a discrete point along the plasma membrane, etc.) or by the creation or destruction of a bond between a receptor and gp120. For example, a single CD4 protein with no neighboring proteins on the plasma membrane would offer four possible new states based on its physical location because of the Cartesian coordinates used to define the flexible plasma membrane. In this example,
is an experimentally determined diffusion coefficient for CD4 on a cellular membrane,
is the three-dimensional distance between the current position of CD4 and that of each available discrete position within the plasma membrane. For an unbound CD4 protein, a new state dictating the movement of the protein would use
. However, an unbound CD4 protein may also offer additional states which do not correspond to the physical movement of the CD4 if it is capable of binding an unbound gp120 protein. Those additional states will vary from the current state by the creation of a previously nonexistent bond; the forward rates toward these states are discussed below. However, should the CD4 protein in question already be participating in an existing bond, the forward rate corresponding to the movement of the bound CD4 protein would have a nonzero
between states. This nonzero
is the change in energy of the existing bond and determines the probability that this CD4 protein moves in an energetically favorable (relaxing the existing bond) or an energetically unfavorable (applying tension or compression to the bond) manner. For the case of receptors located on the plasma membrane, the forward rate constant is calculated using a
determined by the distance between discrete points along the plasma membrane, which vary during the simulation according to the membrane deformation. For system parameters such as the virion position,
is a fixed step size which dictates that the forward rates will vary only according to the change in energy of the existing bonds. Similarly, the transition rates involving a change in z-position of discrete plasma membrane points are calculated using a fixed
. Here the z-position corresponds to the height of each plasma membrane point in the z-axis while the plasma membrane itself is oriented in the x-y plane. A fixed
, dictates that the transition rates of plasma membrane points will vary only according to the local membrane free energy, as discussed below, and if it be the location of a bound cellular receptor, the change in energy of that particular bond.
For the systems examined here, the elapsed time between states and the distance over which physical objects move are of such a small order of magnitude that two assumptions can be made. First, the local energy landscape is approximated to be linear. Second, the probability density is assumed to be at a local steady state. Therefore, at small length scales, the system of the virion and plasma membrane is well described by the high-friction limit of the Fokker-Plank equation.
The on and off rates (
) for CD4 and CCR5 bond formation with gp120 were calculated using experimentally measured rates 
, were computed using a model described by Hummer and Szabo 
is the effective diffusion coefficient,
is the molecular spring constant,
is the distance along the free energy well from the minimum to bond rupture, and
is the minimum bond potential energy. This value of
was then used to calculate
for all bonds using the energy relation
For already existing bonds between cell and virion,
values were calculated using the relation
where the bond potential energy,
, was calculated using a parabolic approximation of the Lennard-Jones potential,
is the distance along the energy potential calculated by subtracting the length of the proteins involved in the bond from the shortest distance between the location of the proteins on the plasma membrane and the virion.
The total probability of transitioning out of the current state was equal to the sum of all forward rates for each possible destination state:
The specific destination state of the system was determined by a pseudo random number generator (PRNG). Briefly, once all possible states available to the current state are determined and their forward rates are calculated, the probability (or rate constant) describing the likelihood that the system transitions away from the current state is the total sum of all forward rates,
. To determine which of these possible states is the next destination state, the PRNG yields a random number,
, uniformly distributed between 0 and 1.
is multiplied by
resulting in a random position between 0 and
, which corresponds to a particular adjacent state. The system is subsequently updated, newly available states are determined, and their new forward rate constants are calculated according to the imposed changes (e.g. if a gp120-CD4 bond breaks, new
rates are calculated for the newly free gp120 and CD4 molecules). This process was repeated, updating each new state sequentially. Energy changes that governed the evolution of the system included those of individual gp120-CD4 bonds, gp120-CCR5 bonds, and the deformation of the plasma membrane with a specified elastic modulus and surface tension.
The fluctuations of the membrane, diffusion of the receptors and the diffusion of the virus are described by Fokker-Planck equations. The simulation methodology is described in Atilgan et al. 
. The total free energy of the system,
, is given by
is the free energy of the plasma membrane calculated using the Canham-Helfrich form 
is the mean curvature of the membrane,
is the local area,
is the elastic modulus and
is the surface tension of the membrane. It should be noted that electrostatic interactions between protein pairs not bound together and between the viral and cellular membranes were not included in the computation of the energy for a given state. In addition, we simplified our model by assuming that the concentration of the local actin filament network beneath the cellular membrane is sufficiently low so as to not dictate plasma membrane deformation 
The time elapsed as the system stepped from one state to another was also calculated and used to determine the total time elapsed during the simulation, starting at
0s. The duration of each time step was calculated using the equation
is the time step from the
+1 state and
is the uniformly distributed random number between 0 and 1 provided by the PRNG.
The plasma membrane was initially defined as a completely flat surface. To simulate a more realistic interaction between cell and virion, the plasma membrane was allowed to evolve during the initialization of the system without the ability to form productive bonds with the virion above it. After this brief initialization (2×106
sequential iterations), the simulation of receptor-mediated viral adhesion to the cell surface was allowed to begin. Surface proteins on the plasma membrane were randomly distributed during the initialization according to the PRNG and concentrations of diffusing, unbound proteins were kept constant throughout the simulation. Proteins on the viral surface were either evenly spaced over the entire particle as previously described 
, or randomly distributed according to the PRNG using a random zenith, θ, between 0 and π according to the probability density distribution
, and random azimuth,
, uniformly distributed between 0 and 2π.
Throughout the simulation, proteins had a finite volume, so that other proteins were not allowed to diffuse through one another on either the viral surface or the plasma membrane. In addition, the plasma membrane and virion could not occupy the same space. If a physical obstacle was encountered, the forward rate for that adjacent state was set equal to 0. The actual lengths of gp120, CD4 and CCR5 molecules were also used when calculating bond interaction distances and free energies (). Lastly, gp120 trimers located on the viral surface were capable of binding up to three CD4 molecules and three CCR5 molecules at a time. As stated earlier, CCR5 adhesion to gp120 in contingent on a previously existing gp120-CD4 bond. In our system, gp120 trimers were not allowed to bind CCR5 unless that trimer was already involved in a gp120-CD4 bond. However, the number of CCR5 bonds was not allowed to exceed the number of CD4 bonds formed with a single gp120 spike, i.e. no synergistic effect was imposed throughout a gp120 trimer that would allow a single CD4 adhesion to promote multiple CCR5 bonds. For a more detailed explanation of the modeling algorithm see Text S1
System parameters used to simulate viral adhesion dynamics.