Theoretical Background

This section reviews the relevant statistical mechanics underlying `conformational flooding'. It is certainly not necessary to understand everything in detail here, however, knowledge on how and why things work helps to obtain better results.

We will treat conformational transitions in a general framework, but focus on collective conformational transitions. Mainly due to entropic barriers, these are generally slow in terms of transition rates and occur on time scales above nanoseconds. However, an actual event of barrier crossing may be as fast as a few picoseconds. For a study of these motions one has (a) to search for distinct low-energy conformations, (b) to find reaction paths connecting the conformations in configuration space, and (c) to estimate transition rates or mean transition times. `Conformational flooding' provides a means to carry out tasks (a) and (b). Estimates can be derived for task (c), but here established techniques are more accurate and, therefore, are recommended.

The potential energy landscape of proteins is quite complex, and so is the free energy landscape. To reduce that complexity, we develop an effective, coarse-grained description with an adjustable level of coarse graining. We will proceed in three steps. First, we will introduce the notion of `conformation space' as a subspace in configuration space. On this subspace we will consider a free energy landscape. Second, we will motivate a proper choice of linear collective coordinates. Third, these `conformational coordinates' will serve to construct a substate model in terms of a harmonic free energy.

From a conventional MD-simulation one obtains an ensemble of Sstructures (`snapshots') ${\bf x}_i$, $i=1\ldots S$, where ${\bf x}_i$ denotes the 3N dimensional configuration vector consisting of the Natomic positions of the N atoms within the system selected to be subjected to the flooding forces. This ensemble is used to estimate the configuration space density $\rho^{{\bf x}}({\bf x})$, which characterizes the initial (known) conformational substate.

We can obtain a coarse-grained description of $\rho^{{\bf x}}$ (i.e., of the initial substate) by considering a number m ( $1\leq m\leq 3N$) of `conformational' (typically collective) degrees of freedom $(c_1,\ldots ,c_m)$, which are assumed to be involved in the conformational motion. Here, m enters as an adjustable parameter, as does the selection of the N atoms to be affected during the flooding simulation.

The m conformational coordinates are extracted from an unperturbed MD simulation by means of a principal component analysis [16]. Below, the averages $\langle\ldots\rangle$ denote ensemble averages over the unperturbed MD trajectory.

From the covariance matrix ${\bf C}:=\langle ({\bf x}-\langle{\bf x}\rangle)({\bf x}-\langle{\bf x}\rangle)^T\rangle$we derive a symmetric, positive definite $3N\times 3N$-matrix ${\bf A}^{-1}:={\bf C}$, which serves to approximate the configuration space density $\rho^{{\bf x}}$in terms of a multivariate Gaussian distribution,  $$\rho^{{\bf x}}({\bf x})\approx Z^{-1} \exp[-\frac{1}{2}({\bf ... ...gle{\bf x}\rangle)^T{\bf A}({\bf x}-\langle{\bf x}\rangle)]\,,$$
where Z is an appropriate normalization.

Care has to be taken that rigid body motions (rotations and translations) are prohibited during that process, since these would `smear out' the configuration space density.

Diagonalizing ${\bf A}={\bf Q}^T{\bf\Lambda}{\bf Q}$ with orthonormal ${\bf Q}\in{\cal R}^{3N\times 3N}$ and diagonal ${\bf\Lambda}=(\delta_{ij}\lambda_i)_{i,j=1,\ldots,3N}$ yields collective coordinates ${\bf q}={\bf Q}({\bf x}-\langle{\bf x}\rangle)$, which serve to simplify Eq. (6.13):  $$\rho^{{\bf x}}({\bf q})\approx Z^{-1}\exp[-\frac{1}{2}{\bf q}^T{\bf\Lambda}{\bf q}]\,.$$
For our coarse-grained description we select the mcollective coordinates ${\bf c}=(q_1,\ldots,q_m)^T$ with smallest eigenvalues $\lambda_i$. That number m of conformational degrees of freedom ${\bf c}_i$which are explicitly considered determines the level of coarse-graining.

On the subspace of the c_i we define a `conformation space density' $\rho^{{\bf c}}({\bf c})$ as the projected configuration space density

$$\begin{eqnarray}\html{eqn34}\rho^{{\bf c}}({\bf c})&=& \int d^{3N}x'\,\rho^{{\bf... ...approx &Z^{-1}\exp[-\frac{1}{2}{\bf c}^T{\bf\Lambda}_c{\bf c}]\,, \end{eqnarray}$$

from which we derive a free energy landscape $F({\bf c})$,

$$\begin{eqnarray}\html{eqn36}F({\bf c})&=&-k_BT\ln\rho^{{\bf c}}({\bf c})\\ &\approx &\frac{1}{2}k_BT{\bf c}^T{\bf\Lambda}_c{\bf c}\,, \end{eqnarray}$$

where k_B is the Boltzmann constant and T the temperature. The latter result is a harmonic approximation, which is used as a model for the initial substate. Note that this harmonic approximation of the free energy landscape differs from the harmonic approximation of the potential energy, which is employed in normal mode analysis. The main difference is that the former includes entropic contributions, whereas the latter does not.

The coarse-grained substate model in terms of $F({\bf c})$ serves to design a `flooding'-potential $V_{\rm fl}({\bf c})$, which is to be included into the force field during MD-simulations, and which is supposed to accelerate conformational transitions. In agreement with our assumption that the conformational coordinates c_i describe conformational transitions sufficiently accurate, we define the flooding potential as a function of only these m degrees of freedom. Note that during a flooding simulation all (not only the mc_i) degrees of freedom are considered, so no real elimination of degrees of freedom takes place here.

Qualitatively, we modify the free energy landscape F as indicated in Fig. 6.3. In the figure, the bold line represents $F({\bf c})$ in the vicinity of a substate (well) as a function of one particular c_i. Also shown is a free energy barrier separating the initial substate from other substates (which are not shown). The purpose of the flooding potential $V_{\rm fl}$ is to rise the free energy within the substate (thin line) as to destabilize that initial substate and to drive the system into another substate. As is also indicated in the figure, we require $V_{\rm fl}$ to be short-ranged, so that the barrier is unaffected. With that assumption, the free energy barrier height is reduced by a destabilization free energy $\Delta F$ indicated in the figure and defined below, and one expects a corresponding acceleration $\exp(\Delta F/k_bT)$ of conformational transitions. That process is termed `conformational flooding'.

 

To ensure that $V_{\rm fl}$ `fits' into the initial substate (cf. Fig. 6.4) we chose a (multivariate) Gaussian,  $$V_{\rm fl}({\bf c}):=E_{\rm fl}% \exp[-\frac{1}{2}{\bf c}^T{\bf\Lambda}_{\rm c}{\bf c}/\gamma^2]$$
(not to be mixed up with the density model $\rho^{\rm c}({\rm c})$, which only accidentally happens to have the same functional form), where $E_{\rm fl}$ is the strength of the flooding potential, and $\gamma=\sqrt{E_{\rm fl}/k_BT}$ determines the overall extension of the flooding potential. With that choice, $V_{\rm fl}$ reduces the depth of the energy well uniformly without extending much into the high energy regions of conformation space, where barriers are to be expected.

 

We would like to stress that our flooding potential will not push the system towards any preselected destination in configuration space; hence, no bias is included as to which product state the system will move. Rather, the method is likely to follow transition paths of low free energy and thus should identify those neighboring conformational substates, to which also the unperturbed system ( $V_{\rm fl}=0$) would move at much slower time scales.

 

As a rule of thumb, Fig 6.5 provides an upper limit for the expected acceleration factor $\tilde{\alpha}$ for various flooding strengths per degree of freedom, $\epsilon_{\rm fl}=E_{\rm fl}/m$. Also given is the corresponding destabilization free energy $f=\Delta F/m$ per degree of freedom.

For a more detailed description, for estimates of the acceleration factor, and for two sample applications, see [22].