J. Phys. Chem. B 2009, 113, 1517–1521
1517
Kinetics and Statistical Distributions of Single-Molecule Conformational Dynamics Qiang Lu† and Jin Wang*,†,‡ Department of Chemistry, Physics and Applied Mathematics, State UniVersity of New York at Stony Brook, Stony Brook, New York 11790, and State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, People’s Republic of China ReceiVed: October 08, 2008
We developed a coarse-grained yet microscopic detailed model to study the statistical fluctuations of singlemolecule protein conformational dynamics of adenylate kinase. We explored the underlying conformational energy landscape and found that the system has two basins of attractions, open and closed conformations connected by two separate pathways. The kinetics is found to be nonexponential, consistent with singlemolecule conformational dynamics experiments. Furthermore, we found that the statistical distribution of the kinetic times for the conformational transition has a long power law tail, reflecting the exponential density of state of the underlying landscape. We also studied the joint distribution of the two pathways and found memory effects. Protein allosteric behavior has been an important issue in understanding biomolecular function with the associated collective molecular conformational changes and reactions.1 The single-molecule measurements provide direct probes to the conformational dynamics through the long time or multiple short time measurements,2-4 and no ensemble average is needed. However, the theoretical studies on single-molecule conformational dynamics so far have been mostly focused on the qualitative behaviors and therefore are on the phenomenological level without kinetics details.2,5,6 In addition, the all-atom molecular dynamics simulation is still difficult to carry out for the whole kinetic process. There have been increasing efforts recently in understanding the conformational dynamics at the coarse-grained level7-13 and more microscopic detailed models.14 Therefore, we are aiming at exploring microscopic details to study the kinetic and statistical aspect of conformational dynamics. We studied an important biologically relevant molecule, adenylate kinase, whose conformational dynamics is crucial for its function of signal transduction and for which NMR15-17 and single-molecule experiments4 have been carried out. The experiments show that there are two stable states, open and closed. X-ray structures also exist where only one domain is closed. The conformational dynamics between the two is complex and often nonexponential. We studied average kinetics, correlations, statistical distributions of conformational switching times, and joint distributions of the biological pathways to uncover the underlying conformational energy landscapes and bridge the theory with the experiments. We developed here a coarse-grained model of protein conformational dynamics at the residue level. Each residue is represented by a single bead centered on its R-carbon (CR) position. There are two key ingredients in our model. One is the (open and closed conformation) structure-based microscopic two well interaction energy function for conformational switch. The other is the long-range water-mediated interaction. * To whom correspondence should be addressed. E-mail: jin.wang.1@ stonybrook.edu. † State University of New York at Stony Brook. ‡ Chinese Academy of Sciences.
Figure 1. Ribbon diagrams of the crystal structures of ADK; (a) open conformation (4AKE) and (b) the close conformation (1AKE). Panel (c) shows the residue-level microscopic interaction energy function between the two residues.
The conformational change of ADK is between open or holo (pdb: 4AKE)19 and closed or apo (pdb: 1ANK)18 structures, shown in Figure 1a and b respectively. The interaction energy U at a given protein conformation Γ is given as N-1
U(Γ, Γ1, Γ2) )
∑ Kb(bi - b0i)
bonds
N-3
θ1i)2 / (θi - θ2i)2 +
(
cos
N-2
2
φ1i - φ2i 2
)]
2
[ ( ()
∑ Kθ(θi -
angles
∑ Kφ cos φi -
dih nonnat
+
+
∑
|i-j|>3
C rij
12
)
φ1i + φ2i 2 nat
+
∑
Unat(rij)
(1)
|i-j|>3
The first three terms control the bond, angle vibration, and dihedral rotation within four adjacent residues. The parameter b0i is the average of bonds of both conformations Γ1 and Γ2 (native open and close); θ1i (θ2i) and φ1i (φ2i) stand for the corresponding variables at the native open (closed) structure Γ1 (Γ2). For the energy parameters, Kb ) 100.0, Kθ ) 20.0, and Kφ ) 1.0.20,21 The fourth and fifth terms represent the native and non-native interaction energies (repulsion) between two nonadjacent residues, respectively. C ()4.0 Å) parametrizes the excluded volume repulsion between residue pairs. The analytical form of the native interactions inspired by ref 22 is given as
10.1021/jp808923a CCC: $40.75 2009 American Chemical Society Published on Web 01/13/2009
{
1518 J. Phys. Chem. B, Vol. 113, No. 5, 2009 Unat(r) ) 1Z(r)(Z(r) - a)
with Z(r) )
[Y(r) /2] - (rh - r1) n
CY(r)
-B
n
2n
Y(r) - h1
with
Y(r)m + h2
[( ) ( ) ]
2 5
r2 r
12
-6
r2 r
10
{
() r1
k
if r < r1
r
2n
+ 2
with
B ) 1m(r2 - rh) h1 ) h2 )
Lu and Wang
{
Y(r) ) (r - rh)2 if r1 e r < rh 4n(1 + 2) C) r2 - rh
2(m-1)
h(m - 1)(r2 - rh)2 m(h + 2) 2(m - 1)(r2 - rh)
if rh e r < r2
2m
h + 2
if r2 e r
(2)
with m ) 5, k ) 8, and n ) 1, so that a smooth and continuous curve can be obtained, as shown in Figure 1c. The first well represents the local short-range contacts based on the native closed conformation structures, while the second well represents the long-range water-mediated interactions caused by many hydrophilic or charged residues among CORE, LID and NMP domain interfaces, and native structure constraints of open conformation from the backbone rigidity. Water molecules can bridge two hydrophilic or charged residues separated by relatively large distances.23 If the two wells are too adjacent (
