Computing Protein−Protein Association Affinity with Hybrid Steered Molecular Dynamics

Abstract

Computing protein-protein association affinities is one of the fundamental challenges in computational biophysics/biochemistry. The overwhelming amount of statistics in the phase space of very high dimensions cannot be sufficiently sampled even with today's high-performance computing power. In this paper, we extend a potential of mean force (PMF)-based approach, the hybrid steered molecular dynamics (hSMD) approach we developed for ligand-protein binding, to protein-protein association problems. For a protein complex consisting of two protomers, P1 and P2, we choose m (≥3) segments of P1 whose m centers of mass are to be steered in a chosen direction and n (≥3) segments of P2 whose n centers of mass steered in the opposite direction. The coordinates of these m+n centers constitute a phase space of 3(m+n) dimensions (3(m+n)-D). All the other degrees of freedom of the proteins, ligands, solvents, and solutes are freely subject to the stochastic dynamics of the all-atom model system. Conducting SMD along a line in this phase space, we obtain the 3(m+n)-D PMF difference between two chosen states, one single state in the associated state ensemble and one single state in the dissociated state ensemble. This PMF difference is the first of four contributors to the protein-protein association energy. The second contributor is the 3(m+n−1)-D partial partition in the associated state counting for the rotations and fluctuations of the (m+n−1) centers while fixing one of the m+n centers of the P1-P2 complex. The two other contributors are the 3(m−1)-D partial partition of P1 and the 3(n−1)-D partial partition of P2 counting for the rotations and fluctuations of their m−1 or n−1 centers while fixing one of the m/n centers of P1/P2 in the dissociated state. Each of these three partial partitions can be factored exactly into a 6-D partial partition multiplying a remaining factor counting for the small fluctuations while fixing three of the centers of P1, P2, or the P1-P2 complex, respectively. These small fluctuations can be well approximated as Gaussian. And every 6-D partition can be reduced in an exact manner to three problems of “1-D sampling”, counting the rotations and fluctuations around one of the centers being fixed. We implement this hSMD approach to the Ras-RalGDS complex, choosing three centers on RalGDS and three on Ras (m=n=3). At a computing cost of about 71.6 wall-clock hours using 400 computing cores in parallel, we obtained the association energy, - 9.2±1.9 kcal/mol on the basis of CHARMM 36 parameters, which well agrees with the experimental data, - 8.4±0.2 kcal/mol.

INTRODUCTION

Accurately computing the free-energy of binding proteins to proteins or ligands is a task of essential importance in biochemical and biophysical studies that still represents considerable challenge to us even with today's high performance computing power.^1–22 An effective approach in the current literature is to use the relationship^{2, 5, 15, 23} between the potential of mean force (PMF)^24–28 and the binding affinity. These PMF-based and other equilibrium sampling approaches have a crucial reliance upon delicate choices of biasing/constraining potentials during the simulation processes and careful removal of the artifacts caused by these artificial potentials. The nonequilibrium steered molecular dynamics (SMD)^29–48 approach, brute force in a certain sense, can be very efficient in sampling forced transition paths from the bound state to the dissociated state but has not been used reliably for free-energy calculations with quantitative accuracy⁴⁶ except the recent development of the hybrid steered molecular dynamics (hSMD) approach for ligand-protein binding problems.⁴⁸

In this further development of the hSMD, we derive a formulation for the protein-protein association affinity of a protein complex consisting two protomers P1 and P2 associated by non-covalent interactions between P1 and P2. This hSMD approach is based on the relationship between the PMF and the binding affinity in the established literature. The widely used SMD involves pulling one center of mass of one selection of the ligand atoms using a spring of finite, carefully chosen, stiffness. In contrast, this hSMD approach involves pulling m (m=3, 4, …) centers of mass of m selected segments of P1 (using m springs of infinite stiffness to disallow any fluctuations of the pulling centers along the way) and likewise (but in the opposite direction) pulling n (n=3, 4, …) centers of mass of n selected segments of P2 to produce a 3(m+n)-dimensional (3(m+n)-D) PMF curve leading from the associated state to the dissociated state of the P1-P2 complex. This PMF difference between the associated state and the dissociated state gives a large (but not dominant) part of the absolute association free energy. Three additional parts represent the rotations and fluctuations of the protomers individually in the dissociated state and of the complex in the associated state. The second part is from the 3(m+n−1)-D partial partition in the associated state counting for the rotation and fluctuations of the (m+n−1) centers while fixing one of the m+n centers of the P1-P2 complex. The two other parts are from the 3(m−1)-D partial partition of P1 and the 3(n−1)-D partial partition of P2 counting for the rotations and fluctuations of their m−1 or n−1 centers while fixing one of the m/n centers of P1/P2 in the dissociated state. Each of these three partial partitions will be shown to factor exactly into a 6-D partial partition in multiplication with a remaining factor. That remaining factor counts for the small fluctuations while fixing three of the centers of P1, P2, or the P1-P2 complex, respectively. These small fluctuations can be well approximated as Gaussian. And every 6-D partition can be reduced in an exact manner to three problems of “1-D sampling”, counting the rotations and fluctuations around one of the centers being fixed.

We carry out applications of this hSMD approach to the Ras-RalGDS complex whose association affinity was experimentally measured and whose structure was determined (PDB code: 1LFD). The computing time required was 71.6 wall-clock hours (all-atom model of 96253 atoms), using 40 Intel Xeon E5-2680 v2 Ivy Bridge 2.8 GHz processors (400 cores) in parallel. The computed absolute free energy of association agrees well with the experimental data.

METHODS

The association affinity/energy from the 3(m+n)-D PMF

Following the established literature,^{2, 5, 15} the association affinity of a dimer protein (consisting of protomers P1 and P2) is related to the 6-D PMF as follows:

$$\frac{1}{k_D∕C_0}=\frac{c_0{\int }_{\text{site}}{d}^3{x}_1^{\left(\mathrm{P}2\right)}\exp \left[-W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_1^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}{\exp {\left[-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}_{\text{bulk}}}$$ (1)

where c₀ is the standard concentration. For clarity and for convenience of unit conversion, we use two different but equivalent forms, c₀=1M on the left hand side and c₀=6.02×10⁻⁴/ų on the right hand side of the equation. k_B is the Boltzmann constant and T is the absolute temperature. The 3-D integrations $(d^3{x}_1^{\left(\mathrm{P}2\right)}\equiv d{\mathbf{r}}_1^{\left(\mathrm{P}2\right)})$ are over the x-, y-, and z-coordinates of the position ${\mathbf{r}}_1^{\left(\mathrm{P}2\right)}$ of protomer P2 that can be chosen as the center of mass of one segment of or the whole protomer P2. Likewise, the position ${\mathbf{r}}_1^{\left(\mathrm{P}1\right)}$ of protomer P1 can be chosen as the center of mass of one segment of or the whole protomer P1. In general, ${\mathbf{r}}_1^{\left(\mathrm{P}1\right)}$ can be fixed at an arbitrarily chosen ${\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)}$ that will be chosen as a point near the center of mass of our model system. The integral has the units of ų that renders the right hand side dimensionless as it should be. $W[{\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_1^{\left(\mathrm{P}2\right)}]$ is the 6-D PMF that is a function of the positions of the protomers P1 and P2. The subscripts “site” and “bulk” indicate, respectively, that ${\mathbf{r}}_1^{\left(\mathrm{P}2\right)}$ is near the PMF minimum and that ${\mathbf{r}}_1^{\left(\mathrm{P}2\right)}={\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)}$ is in the bulk region where the two protomers P1 and P2 are far away from one another.

Since the size of protomer P1 (or P2) is not small and the shape is not simple, the position of one segment center ${\mathbf{r}}_1^{\left(\mathrm{P}1\right)}$ (or ${\mathbf{r}}_1^{\left(\mathrm{P}2\right)}$) will not be sufficient/efficient to represent its location and situation. Instead, the protein complex can be better described with the positions $({\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)})$ of m+n centers of mass of its m+n chosen segments. Fig. 1 shows an example of m=n=3 where the positions $({\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},{\mathbf{r}}_3^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},{\mathbf{r}}_3^{\left(\mathrm{P}2\right)})$ of the six alpha carbons (three of P1 and three of P2) are chosen to quantify the location and situation of the complex. In the associated state, the m+n−1 positions fluctuate without being anyhow biased/constrained during the equilibrium MD simulation while ${\mathbf{r}}_1^{\left(\mathrm{P}1\right)}$ is fixed at ${\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)}$. They are steered during the SMD runs from the bound state to the dissociated state for constructing the 3(m+n)-D PMF $W[{\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]$ as a function of these positions. In the dissociated state, one of the m centers on protomer P1 and one on P2 will be fixed at ${\mathbf{r}}_1^{\left(\mathrm{P}1\right)}={\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)}$ and ${\mathbf{r}}_1^{\left(\mathrm{P}2\right)}={\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)}$ while all others $({\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)})$ rotate and fluctuate according the stochastic dynamics of the system without any other bias/constraint.

Fig. 1.

The three alpha carbons of protomer P1 (RalGDS) and three alpha carbons of P2 (Ras) are shown as black balls marked with their position vectors. The proteins are shown as ribbons colored by residue types. The coordinates are taken from the crystallographic structure (PDB code: 1LFD, Chains A and B). All graphics of this paper were rendered with VMD.⁴⁹

Note that in the relationship between the 6-D and 3(m+n)-D PMFs,

$$\exp \left[-W[{\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_1^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]=C\int \prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=2}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\times \exp \left[-W[{\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]∕k_{B}T\right],$$ (2)

the 3(m+n−2)-D integration over the (m+n−2) positions $({\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)})$ is effectively in a defined neighborhood of $({\mathbf{r}}_1^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)})$ because the protomers each as one whole molecule dictate that the m (or n) centers cannot be stretched much farther away from one another than its molecular size. When ${\mathbf{r}}_1^{\left(\mathrm{P}2\right)}$ is near the binding site, so will be $({\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)})$. When the complex is in the dissociated state, ${\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)}$ of P2 needs to be so far away from the other protomer P1 that integration over $({\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)})$ will be all in the region far from P1. C is the normalization constant that will be cancelled out in the following expressions.

Making use of Eq. (2) twice in Eq. (1) (for the binding site and for the bulk), one has the following expression for the binding affinity,

$$\frac{c_0}{k_D}=\frac{c_0{\int }_{\text{site}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=1}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}{{\int }_{\text{bulk}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=2}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}.$$ (3)

Now inserting the Boltzmann factor at a single state $({\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)})$ chosen from the bound state ensemble and the Boltzmann factor at the corresponding dissociated state ($({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)})$), the binding affinity can be expressed as three contributing factors: The partial partition function at the binding site Z_m−1+n0 of the protein complex, the PMF difference between two chosen states $({\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)})$ and $({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)})$, and the partial partition function in the dissociated state Z_{m−1+n−1∞}. Mathematically,

$$\begin{gathered} \frac{c_0}{k_D}=\frac{c_0{\int }_{\text{site}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=1}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}{\exp \left[-W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]} \\ \times \frac{\exp \left[-W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}{\exp \left[-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}\times \\ \frac{\exp \left[-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]}{{\int }_{\text{bulk}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=2}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]∕k_{B}T\right]} \\ =\frac{c_0{Z}_{m-1+n0}}{Z_{m-1+n-1\infty }}\exp \left[\frac{-\Delta W_{0,\infty }}{K_{B}T}\right]. \end{gathered}$$ (4)

Throughout the text, the subscript 0 refers to the associated state and ∞ refers to the dissociated state. The notation Z_m−1+n0 refers to the fact that in the associated state one of the m pulling centers on P1 is held fixed. Thus the integration is over m−1+n variables. Likewise, the notation Z_{m−1+n−1∞} refers to the fact that, in the dissociated state, P1 and P2 each has one of the pulling centers fixed. Thus the integration is over m+n−2 variables. Here $({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)})$ can be connected to $({\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)})$ via many curves in the 3(m+n)-D space but the PMF is a function of state; thus computation of the PMF difference between the two states can be achieved along a single curve passing through them both. The 3(m+n)-D PMF difference

$$\Delta W_{0,\infty }=W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)}]-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)}]$$ (5)

is between one chosen bound state and its corresponding dissociated state. This PMF difference can be computed by means of the SMD simulations described in the latter part of this section. Note that the one chosen position of the ligand in the bound state,

$$({\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)})=(x_{10}^{\left(\mathrm{P}1\right)},y_{10}^{\left(\mathrm{P}1\right)},z_{10}^{\left(\mathrm{P}1\right)},\dots ,x_{m0}^{\left(\mathrm{P}1\right)},y_{m0}^{\left(\mathrm{P}1\right)},z_{m0}^{\left(\mathrm{P}1\right)};x_{10}^{\left(\mathrm{P}2\right)},y_{10}^{\left(\mathrm{P}2\right)},z_{10}^{\left(\mathrm{P}2\right)},\dots ,x_{n0}^{\left(\mathrm{P}2\right)},y_{n0}^{\left(\mathrm{P}2\right)},z_{n0}^{\left(\mathrm{P}2\right)})$$ (6)

is the starting point for SMD runs. It is taken from the bound state ensemble of the system. It does not have to be the minimum of the PMF but any one state in its close neighborhood. Note that we take the collection of coordinate vectors, e.g., Eq. (6), as a single-row 1 × 3(m + n) matrix. The one state chosen from the dissociated state ensemble

$$({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\mathrm{P}1)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)})=(x_{1\infty }^{\left(\mathrm{P}1\right)},y_{1\infty }^{\left(\mathrm{P}1\right)},z_{1\infty }^{\left(\mathrm{P}1\right)},\dots ,x_{m\infty }^{\left(\mathrm{P}1\right)},y_{m\infty }^{\left(\mathrm{P}1\right)},z_{m\infty }^{\left(\mathrm{P}1\right)};x_{1\infty }^{\left(\mathrm{P}2\right)},y_{1\infty }^{\left(\mathrm{P}2\right)},z_{1\infty }^{\left(\mathrm{P}2\right)},\dots ,x_{n\infty }^{\left(\mathrm{P}2\right)},y_{n\infty }^{\left(\mathrm{P}2\right)},z_{n\infty }^{\left(\mathrm{P}2\right)})$$ (7)

is related to the SMD starting point by a large enough displacement in the 3(m+n)-D space,

$$({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)})=({\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)}-{\mathbf{v}}_{\mathrm{d}}t,\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)}-{\mathbf{v}}_{\mathrm{d}}t;{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)}+{\mathbf{v}}_{\mathrm{d}}t,\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)}+{\mathbf{v}}_{\mathrm{d}}t).$$ (8)

Here v_d is the constant velocity of the SMD pulling and t is the time it takes to steer/pull the P1-P2 complex apart from the associated state to the dissociated state.

As noted above, the partial partition function Z_m−1+n0 of the bound state has the integration over m+n−1 centers and thus has the units of Å^3(m+n−1),

$$\begin{gathered} Z_{m-1+n0}={\int }_{\text{site}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=1}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-\left(W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}] \\ -W[{\mathbf{r}}_{10}^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m0}^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{10}^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{20}^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n0}^{\left(\mathrm{P}2\right)}]\right)∕k_{B}T\right]. \end{gathered}$$ (9)

The partial partition function Z_{m−1+n−1∞} of the dissociated state has the integration over m+n−2 centers and thus has the units of Å^3(m+n−2),

$$\begin{gathered} Z_{m-1+n-1\infty }={\int }_{\text{bulk}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\prod _{i=2}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-\left(W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}] \\ -W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)};{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)}]\right)∕k_{B}T\right]. \end{gathered}$$ (10)

It should be noted that, in the dissociated state, the two protomers are not interacting with each other. Therefore the partial partition function can be factored into two separate partial partition functions of the independent protomers, $Z_{m-1+n-1\infty }=Z_{m-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{n-1\infty }^{\left(\mathrm{P}2\right)}$,

$$\begin{array}{cc}\hfill & Z_{m-1\infty }^{\left(\mathrm{P}1\right)}={\int }_{\text{bulk}}\prod _{i=2}^m{d}^3{x}_i^{\left(\mathrm{P}1\right)}\exp \left[-\left(W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)}]-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)}]\right)∕k_{B}T\right],\hfill \\ \hfill & Z_{n-1\infty }^{\left(\mathrm{P}2\right)}={\int }_{\text{bulk}}\prod _{i=2}^n{d}^3{x}_i^{\left(\mathrm{P}2\right)}\exp \left[-\left(W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}]-W[{\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_{n\infty }^{\left(\mathrm{P}2\right)}]\right)∕k_{B}T\right].\hfill \end{array}$$ (11)

$Z_{m-1\infty }^{\left(\mathrm{P}1\right)}$ refers to the partition function of the m segment centers of P1 in the dissociated state, taking into account explicitly the fact that one of the centers is fixed. $Z_{n-1\infty }^{\left(\mathrm{P}2\right)}$ is the corresponding partition function for P2 in the dissociated state. Again, the use of c₀ = 6.02×10⁻⁴ / ų on the right hand side of Eq. (3) renders it a pure number as desired. And the dissociation constant will conveniently be in the unit of M=mol/L. In summary, we have the following formulas for the association affinity and the absolute free energy of association,

$$\begin{array}{cc}\hfill \frac{c_0}{k_D}& =\frac{c_0{Z}_{m-1+n0}}{Z_{m-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{n-1\infty }^{\left(\mathrm{P}2\right)}}\exp \left[-\frac{\Delta W_{0,\infty }}{k_{B}T}\right],\hfill \\ \hfill \Delta G_{\text{hSMD}}& =k_{B}T\ln \left[\frac{Z_{m-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{n-1\infty }^{\left(\mathrm{P}2\right)}}{Z_{m-1+n0}{c}_0}\right]+\Delta W_{0,\infty }.\hfill \end{array}$$ (12)

In order to determine the association affinity/energy, one needs to compute four factors: (a) The two independent partial partition functions in the dissociated state (Eq. (11)) that are 3(m−1)-D and 3(n−1)-D respectively. (b) The 3(m+n−1)-D partial partition function in the associated state (Eq. (9)). (c) The PMF difference (Eq. (5)) between one chosen bound state and its corresponding dissociated state that can be computed by running SMD simulations of pulling the two protomers forward and backward along a 3(m+n)-D line connecting the associated and the dissociated states. Note that the PMF is a function of state (a point in the 3(m+n)-D space) and the PMF difference is independent of the paths connecting the two end states.

Partial partitions in the dissociated state

As detailed in the Supplemental Information, Section I, each of the two dissociated state partitions can be factored into a 6-D partition and a 3(m−3)-D (3(n−3)-D) partition

$$Z_{m-1\infty }^{\left(\mathrm{P}1\right)}=Z_{3-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{m-3\infty }^{\left(\mathrm{P}1\right)},Z_{n-1\infty }^{\left(\mathrm{P}2\right)}=Z_{3-1\infty }^{\left(\mathrm{P}2\right)}{Z}_{n-3\infty }^{\left(\mathrm{P}2\right)}.$$ (13)

Z_3–1∞ refers to the partial partition function in the dissociated state of three centers with the first center fixed. Z_m–3∞ and Z_n–3∞ are the partial partition functions of the m−3 (n−3) centers with the first three centers are fixed. Naturally, when m=3, $Z_{3-3\infty }^{\left(\mathrm{P}1\right)}=1$ and, when n=3, $Z_{3-3\infty }^{\left(\mathrm{P}2\right)}=1$. When m>3 and/or n>3, $Z_{m-3\infty }^{\left(\mathrm{P}1\right)}$ and $Z_{n-3\infty }^{\left(\mathrm{P}2\right)}$ can be well approximated as Gaussian because, when three centers of a protomer are fixed in the dissociated state, the location and orientation of the promoter will not change much but fluctuate according to the stochastic dynamics. Since the computations of the two partial partitions are identical, we work on protomer P1 and simply replace the superscript P1 with P2 and m with n in the formulas for protomer P2. Fixing the first three pulling centers of P1 at $({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{3\infty }^{\left(\mathrm{P}1\right)})$, the small fluctuations of the other (m−3) centers can be readily sampled to give

$${\mathrm{Z}}_{m-3\infty }^{\left(\mathrm{P}1\right)}={\left(2\pi \right)}^{3(m-3)∕2}{\mathit{Det}}^{1∕2}\left({\Sigma }_{m-3\infty }^{\left(\mathrm{P}1\right)}\right)\exp \left[{\Delta }_{m-3\infty }^{\left(\mathrm{P}1\right)}∕k_{B}T\right].$$ (14)

Here the dimensionless quantity ${\Delta }_{m-3\infty }^{\left(\mathrm{P}1\right)}∕k_{B}T$ gives a measure of how far $({\mathbf{r}}_{4\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{5\infty }^{\left(\mathrm{P}1\right)},\cdots ,{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)})$, the final state of SMD, is from the PMF minimum $(\langle {\mathbf{r}}_4^{\left(\mathrm{P}1\right)}\rangle ,\langle {\mathbf{r}}_5^{\left(\mathrm{P}1\right)}\rangle ,\cdots ,\langle {\mathbf{r}}_m^{\left(\mathrm{P}1\right)}\rangle )$ of P1 in the dissociated state,

$${\Delta }_{m-3\infty }^{\left(\mathrm{P}1\right)}∕k_{B}T=\frac{1}{2}\left(\langle {\mathbf{r}}_4^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{4\infty }^{\left(\mathrm{P}1\right)},\langle {\mathbf{r}}_5^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{5\infty }^{\left(\mathrm{P}1\right)},\dots ,\langle {\mathbf{r}}_m^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)}\right){\left({\Sigma }_{m-3\infty }^{\left(\mathrm{P}1\right)}\right)}^{-1}\times {\left(\langle {\mathbf{r}}_4^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{4\infty }^{\left(\mathrm{P}1\right)},\langle {\mathbf{r}}_5^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{5\infty }^{\left(\mathrm{P}1\right)},\dots ,\langle {\mathbf{r}}_m^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)}\right)}^T.$$ (15)

Det represents the determinant. ${\Sigma }_{m-3\infty }^{\left(\mathrm{P}1\right)}$ is the 3(m−3)×3(m−3) matrix of the fluctuations/deviations of the pulling center coordinates $\delta x_4^{\left(\mathrm{P}1\right)}=x_4^{\left(\mathrm{P}1\right)}-\langle x_{4\infty }^{\left(\mathrm{P}1\right)}\rangle$ etc.,

$$\begin{gathered} {\Sigma }_{m-3\infty }^{\left(\mathrm{P}1\right)}=\langle {\left(\langle {\mathbf{r}}_4^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{4\infty }^{\left(\mathrm{P}1\right)},\langle {\mathbf{r}}_5^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{5\infty }^{\left(\mathrm{P}1\right)},\dots ,\langle {\mathbf{r}}_m^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)}\right)}^T \\ \times \left(\langle {\mathbf{r}}_4^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{4\infty }^{\left(\mathrm{P}1\right)},\langle {\mathbf{r}}_5^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{5\infty }^{\left(\mathrm{P}1\right)},\dots ,\langle {\mathbf{r}}_m^{\left(\mathrm{P}1\right)}\rangle -{\mathbf{r}}_{m\infty }^{\left(\mathrm{P}1\right)}\right)\rangle . \end{gathered}$$ (16)

The brackets 〈(·)〉 stand for the conditional statistical average over the dissociated state ensemble with three of m centers being fixed at $({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{3\infty }^{\left(\mathrm{P}1\right)})$. T means transposing a single-row, 1 × 3(m − 3) matrix into a single-column, 3(m − 3) × 1 matrix. Practically, ${\Sigma }_{m-3\infty }^{\left(\mathrm{P}1\right)}$ can be accurately evaluated by running equilibrium MD in the dissociated state of a protomer while three of m centers are fixed at $({\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{2\infty }^{\left(\mathrm{P}1\right)},{\mathbf{r}}_{3\infty }^{\left(\mathrm{P}1\right)})$. Replacing P1 with P2 and m with n, we have explicitly the 3(n−3)-D partial partition of P2 along with the 3(m−3)-D partial partition of P1

$$\begin{array}{cc}\hfill {\mathrm{Z}}_{m-3\infty }^{\left(\mathrm{P}1\right)}& ={\left(2\pi \right)}^{3(m-3)∕2}{\mathit{Det}}^{1∕2}\left({\Sigma }_{m-3\infty }^{\left(\mathrm{P}1\right)}\right)\exp \left[{\Delta }_{m-3\infty }^{\left(\mathrm{P}1\right)}∕k_{B}T\right],\hfill \\ \hfill {\mathrm{Z}}_{n-3\infty }^{\left(\mathrm{P}2\right)}& ={\left(2\pi \right)}^{3(\mathrm{n}-3)∕2}{\mathit{Det}}^{1∕2}\left({\Sigma }_{n-3\infty }^{\left(\mathrm{P}2\right)}\right)\exp \left[{\Delta }_{n-3\infty }^{\left(\mathrm{P}2\right)}∕k_{B}T\right],\hfill \end{array}$$ (17)

which constitute half of the computations required for the dissociated-state partial partitions in Eq. (13). The other half required in Eq. (13), the 6-D partial partitions can be reduced into 3-D because each protomer's environment in the dissociated state is spherically symmetrical around ${\mathbf{r}}_{1\infty }^{\left(\mathrm{P}1\right)}$ or ${\mathbf{r}}_{1\infty }^{\left(\mathrm{P}2\right)}$. As shown in the Supplemental Information, the 3-D partition can be factored into three 1-D sampling problems in an exact manner,

$$Z_{3-1\infty }^{\left(\mathrm{P}1\right)}=\frac{8{\pi }^2{\left[r_{21\infty }^{\left(\mathrm{P}1\right)}\right]}^2{\left[r_{31\infty }^{\left(\mathrm{P}1\right)}\right]}^2\sin {\theta }_{\infty }^{\left(\mathrm{P}1\right)}}{{\rho }_{\infty }\left(r_{21\infty }^{\left(\mathrm{P}1\right)}\right){\rho }_{\infty }\left(r_{31\infty }^{\left(\mathrm{P}1\right)}\right){\rho }_{\infty }\left({\theta }_{\infty }^{\left(\mathrm{P}1\right)}\right)},Z_{3-1\infty }^{\left(\mathrm{P}2\right)}=\frac{8{\pi }^2{\left[r_{21\infty }^{\left(\mathrm{P}2\right)}\right]}^2{\left[r_{31\infty }^{\left(\mathrm{P}2\right)}\right]}^2\sin {\theta }_{\infty }^{\left(\mathrm{P}2\right)}}{{\rho }_{\infty }\left(r_{21\infty }^{\left(\mathrm{P}2\right)}\right){\rho }_{\infty }\left(r_{31\infty }^{\left(\mathrm{P}2\right)}\right){\rho }_{\infty }\left({\theta }_{\infty }^{\left(\mathrm{P}2\right)}\right)}.$$ (18)

In the rest of this subsection, we imply the superscript (P1) for protomer P1 or (P2) for promoter P2 to save space without causing confusion because the two formulas in Eq. (18) have identical dimensions and structures. The factor of 8π² = 2π × 4π counts for the azimuthal symmetry around an axis (r_2∞ − r_1∞) and the spherical symmetry around a point r_1∞. r₂₁ = |r₂ − r_1∞| and r₃₁ = |r₃ − r_1∞| are the distances between the two pulling centers. θ is the angle between the two vectors r₂ − r_1∞ and r₃ − r_1∞. r_21∞ = |r_2∞ − r_1∞| and r_31∞ = |r_3∞ − r_1∞| are the distances between the fixed centers. θ_∞ is the angle between r_2∞ − r_1∞ and r_3∞ − r_1∞. The three 1-D probability distribution densities are

$$\begin{array}{cc}\hfill {\rho }_{\infty }\left(r_{21}\right)& =r_{21}^2\exp \left[-W_{\infty }\left[r_{21}\right]∕k_{B}T\right]∕\int d{r}_{21}^{\prime }{r}_{21}^{\prime 2}\exp \left[-W_{\infty }\left[r_{21}^{\prime }\right]∕k_{B}T\right],\hfill \\ \hfill {\rho }_{\infty }\left(r_{31}\right)& =r_{31}^2\exp \left[-W_{\infty }\left[r_{31}\right]∕k_{B}T\right]∕\int d{r}_{31}^{\prime }{r}_{31}^{\prime 2}\exp \left[-W_{\infty }\left[r_{31}^{\prime }\right]∕k_{B}T\right],\hfill \\ \hfill {\rho }_{\infty }\left(\theta \right)& =\sin \theta \exp \left[-W_{\infty }\left[\theta \right]∕k_{B}T\right]∕{\int }_0^{\pi }d{\theta }^{\prime }\sin {\theta }^{\prime }\exp \left[-W_{\infty }\left[{\theta }^{\prime }\right]∕k_{B}T\right].\hfill \end{array}$$ (19)

Each of these three constitutes a 1-D sampling problem that can be implemented efficiently to produce accurate results. W_∞[r₂₁] is the 1-D PMF for stretching the protomer between the two pulling centers, steering the second pulling center r₂ to and from the first pulling center that is fixed at r_1∞ along the axis passing through (r_1∞, r_2∞). W_∞[r₃₁] is a function of r₃₁ when the first two centers are fixed at (r_1∞, r_2∞) and the third pulling center r₃ is pulled to and from the first pulling center that is fixed at r_1∞ along the axis passing through (r_1∞, r_3∞). W_∞[θ] is the angular PMF for the third center when the first two centers are fixed at (r_1∞, r_2∞) but the third center r₃ is freely subject to the stochastic dynamics of the system.

Partial partitions in the associated state

In this state, the stochastic dynamics of protomer P1 are tightly coupled to that of P2. It is not necessary to distinguish their coordinates into two separate groups. Instead, one can pick any three out of the (m+n) centers $\left({\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}\right)$ that are denoted as (r₁, r₂, r₃). The rest k=m+n−3 centers are denoted as (r₄, r₅, …, r_m+n). With these notations and the detailed derivation in SI Section II, we have

$$Z_{m-1+n0}=Z_{3-1,0}{Z}_{k0}$$ (20)

Here the 6-D partial partition Z_3–1,0 represents fluctuations of two centers (r₂, r₃) when one center is fixed at r₁ = r₁₀ that can be factored into three 1-D sampling problems in the same manner as Eq. (18),

$$Z_{3-1,0}=\frac{8{\pi }^2{r}_{210}^2{r}_{310}^2\sin {\theta }_0}{{\rho }_0\left(r_{210}\right){\rho }_0\left(r_{310}\right){\rho }_0\left({\theta }_0\right)}.$$ (21)

Noting that the probability distributions here in the associated state take forms similar to those of the dissociated state in Eq. (19),

$$\begin{array}{cc}\hfill {\rho }_0\left(r_{21}\right)& =r_{21}^2\exp \left[-W_0\left[r_{21}\right]∕k_{B}T\right]∕\int d{r}_{21}{r}_{21}^2\exp \left[-W_0\left[r_{21}\right]∕k_{B}T\right],\hfill \\ \hfill {\rho }_0\left(r_{31}\right)& =r_{31}^2\exp \left[-W_0\left[r_{31}\right]∕k_{B}T\right]∕\int d{r}_{31}{r}_{31}^2\exp \left[-W_0\left[r_{31}\right]∕k_{B}T\right],\hfill \\ \hfill {\rho }_0\left(\theta \right)& =\sin \theta \exp \left[-W_0\left[\theta \right]∕k_{B}T\right]∕{\int }_0^{\pi }d\theta \sin \theta \exp \left[-W_0\left[\theta \right]∕k_{B}T\right].\hfill \end{array}$$ (22)

W₀[r₃₁] as a function of r₃₁ = |r₁ − r₃|, the distance between the first and the third center, is the 1-D PMF along the (r₁₀, r₃₀) direction when two centers are fixed at (r₁₀, r₂₀). W₀[r₂₁] as a function of r₂₁ = |r₁ − r₂|, the distance between the first and the second center, is the 1-D PMF along the (r₁₀, r₂₀) direction when only one center is fixed (r₁₀). r₃₁₀ = |r₁₀ − r₃₀| and r₂₁₀ = |r₁₀ − r₂₀| are, respectively, the distance between the 3^rd and the 1^st pulling centers and the same between the 2^nd and the 1^st pulling centers.

Z_k0 represents fluctuations of k centers (r₄, r₅, …, r_m+n) when three center are fixed at (r₁₀, r₂₀, r₃₀). When the protein association is tight, one can approximate the fluctuations as Gaussian in the neighborhood of the PMF minimum. The coordinates of the minimum of a Gaussian distribution are equal to the average coordinates, of course, (〈r₄〉, 〈r₅〉, ⋯, 〈r_m+n〉).

$${\mathrm{Z}}_{k0}={\left(2\pi \right)}^{3k∕2}{\mathit{Det}}^{1∕2}\left({\Sigma }_{ko}\right)\exp \left[{\Delta }_{ko}∕k_{B}T\right].$$ (23)

Here the dimensionless quantity Δ_ko/k_BT gives a measure of how far (r₄₀,r₅₀, ⋯, r_m+n0), the initial state chosen for SMD, is from the PMF minimum (〈r₄〉, 〈r₅〉, ⋯, 〈r_m+n〉),

$${\Delta }_{ko}∕k_{B}T=\frac{1}{2}\left(\langle {\mathbf{r}}_4\rangle -{\mathbf{r}}_{40},\langle {\mathbf{r}}_5\rangle -{\mathbf{r}}_{50},\cdots ,\langle {\mathbf{r}}_{m+n}\rangle -{\mathbf{r}}_{m+n0}\right){\Sigma }_{ko}^{-1}{\left(\langle {\mathbf{r}}_4\rangle -{\mathbf{r}}_{40},\langle {\mathbf{r}}_5\rangle -{\mathbf{r}}_{50},\cdots ,\langle {\mathbf{r}}_{m+n}\rangle -{\mathbf{r}}_{m+n0}\right)}^T.$$ (24)

Σ_ko is the 3k×3k matrix of the fluctuations/deviations of the pulling center coordinates δx₄ = x₄ − 〈x₄〉 etc.

$${\Sigma }_{ko}=\langle {\left(\langle {\mathbf{r}}_4\rangle -{\mathbf{r}}_{40},\langle {\mathbf{r}}_5\rangle -{\mathbf{r}}_{50},\cdots ,\langle {\mathbf{r}}_{m+n}\rangle -{\mathbf{r}}_{m+n0}\right)}^T\left(\langle {\mathbf{r}}_4\rangle -{\mathbf{r}}_{40},\langle {\mathbf{r}}_5\rangle -{\mathbf{r}}_{50},\cdots ,\langle {\mathbf{r}}_{m+n}\rangle -{\mathbf{r}}_{m+n0}\right)\rangle .$$ (25)

${\Sigma }_{ko}^{-1}$ is the inverse matrix of Σ_ko which can be accurately evaluated by running equilibrium MD in the associated state of the protein-protein complex while three of m+n centers are fixed at (r₁₀,r₂₀,r₃₀).

PMF from SMD simulations

In an SMD²⁹ simulation of the current literature, one steers/pulls one center of mass of one selection of atoms, using a spring with a carefully chosen stiffness (spring constant). The use of a spring of finite stiffness introduces additional fluctuation and dissipation in the added degrees of freedom.⁴⁴ In this paper, we choose m+n segments (mutually exclusive m+n selections of atoms) of the protomers P1 and P2 for steering/pulling with m+n infinitely stiff springs (m, n=1, 2, 3……). Namely, the m+n centers of mass of the chosen m+n segments will be controlled as functions of time t

$${\mathbf{r}}_i={\mathbf{r}}_{iA}\pm {\mathbf{v}}_{\mathbf{d}}t,i=1\dots m+n$$ (26)

while all the other degrees of freedom of the system are freely subject to stochastic dynamics. Here r_i = (x_i, y_i, z_i) is the center of mass coordinates of the i-th segments, v_d is the pulling velocity, and r_iA are coordinates of the centers of mass of the steered segments at the end state A. The + and − signs are for the forward and reverse pulling paths, respectively, for protomer P1 and the opposites to these signs for P2. {r_i} denotes ($\left({\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},\dots ,{\mathbf{r}}_m^{\left(\mathrm{P}1\right)};{\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},\dots ,{\mathbf{r}}_n^{\left(\mathrm{P}2\right)}\right)$) etc. We adopt the multi-sectional scheme of Ref. ⁵⁰. The path from the bound state to the dissociated state is divided into a number of sections. Within a given section whose end states are marked as A and B, respectively, multiple forward and reverse pulling paths are sampled along which the work done to the system is recorded. The Gibbs free-energy difference (namely, the PMF or the reversible work) is computed via the Brownian-dynamics fluctuation-dissipation theorem (BD-FDT)⁵¹ as follows:

$$W\left[\left\{{\mathbf{r}}_i\right\}\right]-W\left[\left\{{\mathbf{r}}_{iA}\right\}\right]=-k_{B}T\ln \left(\frac{{\langle \exp [-W_{\left\{{\mathbf{r}}_{iA}\right\}\to \left\{{\mathbf{r}}_i\right\}}∕2k_{B}T]\rangle }_F}{{\langle \exp [-W_{\left\{{\mathbf{r}}_i\right\}\to \left\{{\mathbf{r}}_{iA}\right\}}∕2k_{B}T]\rangle }_R}\right).$$ (27)

Here the brackets with subscript F/R represent the statistical average over the forward/reverse paths. W_{riA}→{ri} is the work done to the system along a forward path when the proteins are steered from A to r. W_{ri}→{riA} = W_{{riB}→{riA}} − W_{riB}→{ri} is the work for the part of a reverse path when the protomers are pulled from r to A. {r_iA}, {r_i}, and {r_iB} are the coordinates of the centers of mass of the steered segments at the end state A, the general state r, and the end state B of the system, respectively. At each end of a section, A/B, the system is equilibrated for a time long enough to reach conditioned equilibrium while the steered centers are fixed at {r_iA}/{r_iB}. In this way, running SMD section by section, we map the PMF W[{r_i}] as a function of the steered centers along a chosen path from the associated state to the dissociated state.

Simulation parameters

In all the equilibrium MD and nonequilibrium SMD runs, we used the CHARMM36^52–53 force field for all intra- and inter-molecular interactions. We implemented Langevin stochastic dynamics with NAMD⁵⁴ to simulate the systems at constant temperature of 298 K and constant pressure of 1 bar. Full electrostatics was implemented by means of particle mesh Ewald (PME) at 128×128×256. The time step was 1 fs for short-range and 2 fs for long-range interactions. The PME was updated every 4 fs. The damping constant was 5/ps. Explicit solvent was represented with the TIP3P model. The pulling was along the ±z-axis at a speed of v_d = (0, 0,±1.25Å/ns) so that the two partners are separated at a speed of 2.5Å/ns in all SMD runs except one test case when the separation speed was chosen as 1.25Å/ns to illustrate the accuracy of the hSMD method. In all sections, four forward and four reverse pulling paths were sampled.

The all-atom model system of Ras-RalGDS complex, shown in Fig. 2, was formed from the crystallographic structure (PDB code: 1LFD)⁵⁵ by taking its A and B chains, rotating it to the orientation of RalGDS on top Ras along the z-axis, replacing GNP with GTP, putting the complex in the center of a water box of 80Å×80Å×160Å, and neutralized with 12 Na⁺ ions. During the 50 ns equilibrium MD run, the system settles down (after 3 ns) to fluctuate slightly around the dimensions of 80Å×80Å×147Å. To correspond exactly with the in vitro experimental conditions,⁵⁶ 50 mM Tris/HCl, 5 mM MgCl₂, and 100 mM NaCl were added to the model system.

Fig. 2.

All-atom model system of the Ras-RalGDS complex. (A) The simulation box of 80Å×80Å×147Å in dimensions. The proteins reside near the center of the water box. Also visible are some of the counter ions. (B) The protein complex in its associated state. (C) The Ras-RalGDS complex (ribbons, colored by residue types) with six steering centers marked: Three alpha carbons of Asn 29, Lys 32, and Lys 52 are shown as gold spheres to be steered along the z-direction. Three alpha carbons of Ile 221, Asp 238, and Lys 242 are shown as purple spheres to be steered along the negative z-direction. (D) Ras and RalGDS in dissociated state. Shown in (B) and (D) are Ras (colored yellow) and RalGDS (colored red) in ribbons, GTP in large spheres (colored by atom names), and the interfacial residues of Ras (in balls and sticks, colored by atom names) and of RalGDS (in licrorices, colored by atom names).

RESULTS

Analytical formula for protein-protein association affinity

The derivation of the previous section is our main result, which is summarized in Table 1 to Table 3.

Table 1.

Summary of dissociation constant calculation.

Quantity	Description	Equation/calculation method
kD	Dissociation constant	$\frac{1}{k_D}=\frac{Z_{m-1+n0}}{Z_{m-1+n-1\infty }}\exp \left[\frac{-\Delta W_{0,\infty }}{k_{B}T}\right]$
Z m−1+n−1∞	Partial partition function of proteins in the dissociated state (one center fixed on each protein)	$Z_{m-1+n-1\infty }=Z_{m-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{n-1\infty }^{\left(\mathrm{P}2\right)}$
Z m−1+n0	Partition function in the associated state (one center fixed on the protein complex)	Zm−1+n0 = Z3−1,0Zk0
Δ W 0,∞	PMF difference between a chosen state in the associated state ensemble and the corresponding dissociated state.	Calculated through SMD, Eq. (27)

Table 3.

Partial partition function of one protein (P1) in the dissociated state. Identical formulas for protein P2 by substitution of P1 with P2.

Z m−1+n−1∞	Partial partition function of proteins in the dissociated state (one center fixed on each protein)	$Z_{m-1+n-1\infty }=Z_{3-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{m-3\infty }^{\left(\mathrm{P}1\right)}{Z}_{3-1\infty }^{\left(\mathrm{P}2\right)}{Z}_{n-3\infty }^{\left(\mathrm{P}2\right)}$
$Z_{3-1\infty }^{\left(\mathrm{P}1\right)}$	Partial partition of P1 in the dissociated state when one of the centers fixed	$Z_{3-1\infty }^{\left(\mathrm{P}1\right)}=\frac{8{\pi }^2{\left[r_{21\infty }^{\left(\mathrm{P}1\right)}\right]}^2{\left[r_{31\infty }^{\left(\mathrm{P}1\right)}\right]}^2\sin {\theta }_{\infty }^{\left(\mathrm{P}1\right)}}{{\rho }_{\infty }\left(r_{21\infty }^{\left(\mathrm{P}1\right)}\right){\rho }_{\infty }\left(r_{31\infty }^{\left(\mathrm{P}1\right)}\right){\rho }_{\infty }\left({\theta }_{\infty }^{\left(\mathrm{P}1\right)}\right)}$
r ij ∞	Distance between ri∞ and rj∞ (positions of the ith and the jth centers in the dissociated state corresponding to the one state chosen from the associated state ensemble)	rij∞ = |ri∞ − rj∞|
θ ∞	Angle between r2∞ − r1∞ and r3∞ − r1∞	$\text{arccos}\left[\frac{({\mathbf{r}}_{2\infty }-{\mathbf{r}}_{1\infty })\cdot ({\mathbf{r}}_{3\infty }-{\mathbf{r}}_{1\infty })}{\mid {\mathbf{r}}_{2\infty }-{\mathbf{r}}_{1\infty }\mid \mid {\mathbf{r}}_{3\infty }-{\mathbf{r}}_{1\infty }\mid }\right]$
ρ∞ (·)	1-D probability distribution densities	Calculated through MD sampling, see Eq. (19)
$Z_{m-3\infty }^{\left(\mathrm{P}1\right)}$	Partial partition of P1 in the dissociated state when three of the m centers fixed	Gaussian approximation, see Eq. (17)

For the ras-ralGDS complex and the barnase-barstar complex (SI, Section V), we choose m=n=3, three pulling centers on each of the two partner proteins and we use Gaussian approximations for the fluctuations in the associated state when three centers are fixed, (23). Then we have, from Eq. (12), the free energy of association as follows:

$$\Delta G_{\text{hSMD}}=\Delta W_{0,\infty }-{\Delta }_{3o}+k_{B}T\ln \left[\frac{Z_{3-1\infty }^{\left(\mathrm{P}1\right)}{Z}_{3-1\infty }^{\left(\mathrm{P}2\right)}}{c_0{\left(2\pi \right)}^{9∕2}{\mathit{Det}}^{1∕2}\left({\Sigma }_{3o}\right)Z_{3-1,0}}\right]$$ (28)

where three 6-D partial partitions are all factored out into 1-D partial partitions as shown Tables 2 and 3. c₀ = 6.02×10⁻⁴/ų. Δ_3o and Σ_3o given in Eq. (24) and Eq. (25), respectively, with k=3.

Table 2.

Partial partition of the associated state.

Z m−1+n0	Partial partition function in the associated state (one center fixed on the complex)	Zm−1+n0 = Z3−1,0Zk0
Z 3−1,0	Partial partition for the fluctuations of the complex in the associated state when one center is fixed	$Z_{3-1,0}=\frac{8{\pi }^2{r}_{210}^2{r}_{310}^2\sin {\theta }_0}{{\rho }_0\left(r_{210}\right){\rho }_0\left(r_{310}\right){\rho }_0\left({\theta }_0\right)}$
r ij0	Distance between ri0 and rj0 (positions of the ith and jth centers in the one state chosen from the associated state ensemble)	rij0 = |ri0 − rj0|
θ 0	Angle between r20 − r10 and r30 − r10	$\text{arccos}\left[\frac{({\mathbf{r}}_{20}-{\mathbf{r}}_{10})\cdot ({\mathbf{r}}_{30}-{\mathbf{r}}_{10})}{\mid {\mathbf{r}}_{20}-{\mathbf{r}}_{10}\mid \mid {\mathbf{r}}_{30}-{\mathbf{r}}_{10}\mid }\right]$
ρ0 (·)	1-D probability distribution densities	Calculated through MD sampling, see Eq. (22)
Z k0	Partial partition for fluctuations of the k=m+n−3 centers in the associated state with the three of the m+n centers fixed	Gaussian approximation, see Eq. (23)

Numerical results

For the Ras-RalGDS system, we choose three alpha carbons on each protein as the steering/pulling centers: ASN 29, LYS 32, and LYS 52 on RalGDS (noted as P1, their position vectors noted as $\left({\mathbf{r}}_1^{\left(\mathrm{P}1\right)},{\mathbf{r}}_2^{\left(\mathrm{P}1\right)},{\mathbf{r}}_3^{\left(\mathrm{P}1\right)}\right)$) along with ASP 238, ILE 221, and LYS 242 on Ras (notated as P2, position vectors as $\left({\mathbf{r}}_1^{\left(\mathrm{P}2\right)},{\mathbf{r}}_2^{\left(\mathrm{P}2\right)},{\mathbf{r}}_3^{\left(\mathrm{P}2\right)}\right)$). See Fig. 1 and Fig. 2(C) for illustrations. The numerical results are tabulated in Table 4. Note that each of the three 6-D partial partitions involved in Eq. (28) was reduced three 1-D sampling problems for which the numerical data were presented in SI Figs. S1 and S2. The 9×9 matrix Σ_3o for fluctuations in the associated state defined in Eq. (25) and the deviations of the initial state from the equilibrium average state Δ_3o as measured in Eq. (24) for k=3 were computed from the data shown in SI, Section III, Fig. S3. The 18-D PMF along the puling path illustrated in Fig. 3 was computed from the work curves shown in SI, Section IV, Fig. S4 using Eq. (27). The PMF difference ΔW_0,∞ was taken between z=0 and z=12Å.

Table 4.

Computed results of ras-ralGDS complex.

The 6-D partial partition of ralGDS in the dissociated state when one center is fixed	$Z_{3-1\infty }^{\left(\mathrm{P}1\right)}$	7.05×105 Å6
The 6-D partial partition of ras in the dissociated state when one center is fixed	$Z_{3-1\infty }^{\left(\mathrm{P}2\right)}$	1.34×106 Å6
The 6-D partial partition of ras-ralGDS complex in the associated state when one center is fixed	Z 3−1,0	7.63×106 Å6
The 9×9 matrix for fluctuations in the associated state defined in Eq. (25)	Det(Σ3o)	7.036×10−12 Å18
The deviations of the initial state from the equilibrium average state as measured in Eq. (24) for k=3	Δ 3o	3.79 kcal/mol
PMF difference between the one chosen associated state and the corresponding dissociated state	Δ W 0,∞	−18.2 kcal/mol
The absolute free energy of ras-ralGDS association, computed with hSMD, in aqueous solution of 50 mM Tris/HCl, 5 mM MgCl2, and 100 mM NaCl.	Δ G hSMD	−9.2±1.9 kcal/mol
The experimental data of the ras-ralGDS association free energy measured by ITC56 in aqueous solution of 50 mM Tris/HCl, 5 mM MgCl2, and 100 mM NaCl.	Δ G ITC	−8.4±0.2 kcal/mol
The ras-ralGDS association free energy, computed by MM/GBSA1, in aqueous solution of 100 mM NaCl.	Δ G GBSA	−19.5±5.9kcal/mol

Fig. 3.

PMF along the pulling path from the associated state to the dissociated state. The horizontal axis indicates displacement of RalGDS relative to Ras when they are steered in the positive and the negative z-direction, respectively. The second set of data were obtained at a slower pulling speed of 1.25 Å/ns to demonstrate convergence of the numerical sampling. Note that equilibrations were only performed at z=0 and z=4 Å but not in between them. Therefore, the convergence is only expected for the PMF difference between the two points, not in between.

DISCUSSION

It is worth noting that the PMF difference (−18.2 kcal/mol) is very far off the computed free energy of association (−9.2 kcal/mol) in this case and for protein-protein associations in general because it is only the difference between two single states (two points in the 18-D space), one state chosen from the associated state ensemble and the corresponding one state in the dissociated state ensemble. The fluctuations around these two single states are represented in the 15-D partial partition (the associated conformation) and the 12-D partial partition (the dissociated conformation), respectively. Shown in the Fig. 4 and in SI, Section III, are the behaviors of these fluctuations. They give equally important contributions (9.0 kcal/mol) as one would expect.

Fig. 4.

RMSD of the proteins from the crystallographic structure (PDB: 1LFD). (A) RalGDS in dissociated state. (B) Ras in dissociated state. (C) Ras-RalGDS complex in associated state.

In particular, we use the crystal structures of Ras and RalGDS as the reference to compute the structural deviations of Ras and RalGDS in associated or dissociated state that are shown in Fig. 4. In the dissociated state, one center is fixed on each protein Ras or RalGDS. In the associated state, only one center is fixed on Ras while RalGDS follows the stochastic dynamics in interactions with Ras. The backbone structure of the Ras-RalGDS complex does not deviate from the crystal structure more than 1.7 Å and not more than 2.2 Å even including the fluctuations of the sidechains (Fig. 4(C)), indicating the stability of Ras-RalGDS complex and the quality of the crystal structure data.⁵⁵ In the dissociated state, Ras does not deviate significantly from the crystal structure (<1.2 Å for backbones) and its sidechains do not fluctuate more than 1.9 Å on the average. In contrast, RalGDS's deviations are as high as 2.3 Å for backbones and 2.8 Å for sidechains. These fluctuations count for most of the part of the association energy on top of the PMF difference.

What interactions are responsible for the Ras-RalGDS association? As shown in Fig. 5, the Ras protein residues in contact with RalGDS are: Lys 231, Asp 233, Pro 234, Thr 235, Ile 236, Glu 237, Asp 238, Ser 239, Tyr 240, Arg 241, Leu 256, Tyr 264, and Met 267. They are in contact in the crystal structure and they remain in close contact during the 50 ns equilibration process, indicating their relevance in the Ras-RalGDS recognition.

Fig. 5.

Proteins Ras (in ribbons, colored by residue types) and RalGDS (in surface, colored by residue types) in associated state viewed from four different angles. Also shown are the Ras residues (in spheres, colored by atom names) in contact with RalGDS. The coordinates were from the end of the 50 ns MD run of the associated state.

Along the forced separation path, the PMF does not rise monotonously (Fig. 3). The peculiarity around 2 to 4 Å reflects the importance of the following interactions: the salt bridges between Glu 237 of Ras-Arg 20 of RalGDS and between Asp 233 of Ras-Lys 52 of RalGDS along with the vdW contacts between Ile 236 of Ras-(Ile 18, Leu 35) of RalGDS. These interactions cause conformational distortions before their separation as shown in Fig. 5. We therefore confirm that they are the main contributors to the Ras-RalGDS binding.^{1, 55}

It should be helpful to note that hSMD is intended to be a “brute force” method that does not require much sophistication of sampling schemes but yet gives estimation of standard binding free energy with chemical accuracy. The PMF is a function of state and consequently the PMF difference between two states can be obtained via multiple paths connecting the two end states of the paths sampled. This fundamental principle enables us to compute the PMF difference accurately by sampling a few forward and the same number of reverse pulling paths connecting one chosen state in the associated state ensemble and its corresponding state in the dissociated state ensemble. Naturally, equilibrium sampling of the associated state ensemble and the same of the dissociated state ensemble must be carried out to account for all the relevant fluctuations of the pulling centers. These 3(k+3)-D sampling problems are resolved as 3k-D Gaussian fluctuations of k centers when three of the pulling centers are fixed in multiplication with 6-D sampling problems. Here k=m+n−3 for the associated P1-P2 complex, k=m−3 for P1 and k=n−3 for P2 in the dissociated state. Therefore the choice of the pulling centers, the k centers in particular, need to be the most stable parts of the proteins so that the fluctuations can be accurately accounted for in the Gaussian approximation. In the two protein-protein complexes studied in this work, we made the choice of three α-carbons on each protein without multiple trials. Given that m=n=3 is the crudest implementation of hSMD, the agreement of our results with the experimental data suggests that hSMD is not very sensitive to the choice of pulling centers. Finer choice of the same number of pulling centers will certainly lead to better accuracy without additional computing cost. Using m>3 and/or n>3 will involve more computing and is expected to give better accuracy in terms of statistical mechanics. All this depends on the accuracy of the force field parameters we use to characterize the intra- and inter-molecular interactions of the systems, of course.

CONCLUSIONS

We have developed a hybrid steered molecular dynamics approach for computing the absolute free energy of protein-protein association from the PMF along a dissociation path. Applying this hSMD approach with high-performance parallel processing, one can obtain, within a few wall-clock days, the association affinity of one protein-protein complex with accuracy comparable with experimental measurements. Using this “brute force” approach, one does not have to delicately devise biasing and constraining potentials during the course of simulations. One simply steers/pulls the m+n centers of mass of m (≥3) chosen segments of one protein and n (≥3) chosen segments of the partner protein by using m+n infinitely stiff springs along one predetermined path of separation, disallowing any deviations from the path. This use of a single path correctly gives the 3(m+n)-D PMF difference between one state in the ensemble of the associated states and its counterpart in the ensemble of the dissociated states, noting that the PMF is a function of state and thus the PMF difference between two states is independent of the paths connecting them. All other contributions, in addition to the PMF difference between the two end states of this one dissociation path, are rigorously accounted for in the 3(m+n−1)-D partial partition function of the associated state and two partial partitions of the dissociated states that are 3(m−1)-D and 3(n−1)-D respectively. Each of the three partial partitions can be well approximated as the product of a 6-D partial partition and the Gaussian fluctuations when three of the centers are fixed and the 6-D partial partition can be factored into three 1-D sampling problems. The total computing cost for our hSMD study of the Ras-RalGDS complex was approximately three wall-clock days using 400 computing cores (20 nodes) on the TACC supercomputer Maverick (no GPU usage) or, based on a prorate estimate, 12 wall-clock days using one node (20 cores and one nVidia K40) with GPU usage. We hereby assert that the hSMD approach is efficient for protein-protein binding affinities given that today's high performance computing power affords us a proportionally large number of computing cores needed for a given protein-protein system.

Studying the Ras-RalGDS complex, we have provided atomistic details in support of the binding mechanisms elucidated in the experimental and theoretical investigations in the current literature. Our equilibrium MD simulations confirm the experimentally determined binding conformations. During the long time dynamics, the protomers were found to fluctuate around the crystal structure coordinates with deviations less than 2 Å. And our computed association energy agrees with the experimentally measured value within the margin of error. We expect that the hSMD method is usable to accurately predict association affinities of other protein-protein complexes.

Fig. 6.

Residues pulled out of their ways along the SMD run. Asp 233, Ile 236 and Glu 237 of Ras are in close contact with Lys 52, (Ile 18 and Leu 35), and Arg 20 of RalGDS, respectively. Ras (yellow) and RalGDS (red) are shown in ribbons and the afore-listed residues are shown in spheres colored by residue types.

Abbreviations

1-D

1-dimensional

m-D

m-dimensional

GBSA

generalized Born surface area

GNP

phosphoaminophosphonic acid guanylate ester

GTP

guanosine-5'-triphosphate

hSMD

hybrid steered molecular dynamics

ITC

isothermal titration calorimetry

k_D

dissociation constant

MD

molecular dynamics

PME

particle mesh Ewald

PMF

potential of mean force

RalGDS

Ral guanine nucleotide dissociation stimulator

RMSD

root mean square deviation

SMD

steered molecular dynamics

VDW

van der Waals