Brownian Dynamics

In this approach the protein was supposed to be in a stochastic bath that keeps the temperature constant and modulates the otherwise extreme oscillations of the residues. This bath was simulated with two terms accounting for a velocity-dependent friction and stochastic forces due to the solvent environment (eq. 1). Velocity Verlet algorithm was used to solve the stochastic differential equation for alpha-carbon shown in eq. 1.

$m\dot{\nu}_i=\gamma \nu_i+F_i+\eta_i$

(1),

where m stands for the effective mass of Cα (see below), $\nu$ and $\dot{\nu}$ stands for velocity and acceleration, $F$ represent the force; $\gamma$ is the inverse of a characteristic time at which the particle loses its energy in a given solvent, and finally the random term is considered a Robust white noise ( $\eta\left (t\right )$ ) with autocorrelation given by eq. 2.

$\left< \eta_l\left( t \right)\eta_n\left( t^\prime \right) \right>=2mk_BT\gamma\delta_{ln}\delta\left( t-t^\prime\right)$

(2),

where $k_B$ is the Boltzmann constant, and $T$ is the temperature of the stochastic bath. The Dirac functions $\delta_ln$ and $\delta\left(t-t^\prime\right)$ forces the independence of the components of the noise vector.

The equation of motion (eq. 1) is integrated using of Verlet’s algorithm, giving for the velocities (eq. 3) and positions (eq. 4) after time $\Delta t$ :

$\vec{\nu_i}=e^{-\frac{\Delta t}{\tau}}\vec{\nu_i^0}+\frac{1}{\gamma}\left( 1-e^{-\frac{\Delta t}{\tau}}\right)\vec{F_i^0}+\Delta\vec{\nu_i^G}$

(3),

and

$\vec{r_i}=\vec{r_i^0}+\tau\left(1-e^{-\frac{\Delta t}{\tau}}\right)\vec{\nu_i^0}+\frac{\Delta t}{\gamma}\left( 1-\frac{\tau}{\Delta t}\left( 1-e^{-\frac{\Delta t}{\tau}}\right)\right)\vec{F_i}+\Delta\vec{r_i^G}$

(4),

where $\tau=m\gamma^{-1}$ is the characteristic time, and $\Delta\vec{r_i^G}$ , $\Delta\vec{\nu_i^G}$ are the changes in position and velocity induced by the stochastic term.

The potential energy used to compute forces in eq. 1 assumes a coarse-grained representation of the protein (Cα-only) and a quasi-harmonic representation of the interactions (eq. 5) similar to that suggested by Kovacs et al. 2004

$U_{ij}=\frac{1}{2}C\left(\frac{r^*}{\left|\vec{r_{ij}^0}\right|}\right)^6\left(\vec{r_{ij}}-\vec{r_{ij}^0}\right)^2$

(5),

where $\vec{r_{ij}}=\vec{r_i}-\vec{r_j}$ stands for the vector connecting Cα atoms i and j.

The initial condition is a native structure (the MD-averaged conformation in this work) that is supposed to be in the minimal energy state, from which the relative vectors $\vec{r_{ij}^0}$ are computed. After some tests, factor C is taken to be 40 kcal/mol Å² and r* is the mean distance between two consecutive Cα atoms set to 3.8Å. The mass of all Cα atoms is set to 100 daltons (i.e, that of an average residue). The velocity-dependent friction $\gamma$ is considered to have the same value as for water (i.e., 0.4 ps^-1). BD simulation time scales were equivalent to those considered in MD.