Local transition gradients determine the global attributes of protein energy landscapes

0.24 MB·English

by  Francesco Rao

Local transition gradients determine the global attributes of protein energy landscapes
Francesco Rao
Laboratoire de Chimie Biophysique/ISIS, Universite de Strasbourg, 8, alle G. Monge, Strasbourg, France
(Dated: January 20, 2010)
The dynamical characterization of proteins is crucial to understand protein function. From a
microscopic point of view, protein dynamics is governed by the local atomic interactions that, in
turn, trigger the functional conformational changes. Unfortunately, the relationship between local
atomicfluctuationsandglobalproteinrearrangementsisstillelusive. Here,atomisticmoleculardy-
namicssimulationsinconjunctionwithcomplexnetworkanalysisshowthatfastpeptiderelaxations
effectively build the backbone of the global free-energy landscape, providing a connection between
localandglobalatomicrearrangements. Aminimum-spanning-treerepresentation,builtonthebase
of transition gradients networks, results in a high resolution mapping of the system dynamics and
thermodynamicswithoutrequiringanyaprioriknowledgeoftherelevantdegreesoffreedom. These
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results suggest the presence of a local mechanism for the high communication efficiency generally
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observed in complex systems.
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2 Keywords: ProteinDynamics,MolecularSimulations,Free-energyLandscapes,ComplexNetworks

n
a
J Incomplexsystemsthebehaviorofthewholeishardly different microstates and whose links correspond to di-
0 predictable from the fundamental laws of interactions of recttransitionsbetweenthem[13–15]. Thisapproachhas
2 the single components [1]. Indeed, it is hard to reveal been successfully applied to the study of peptide folding
the intimate connection between the local properties of and structural transitions [14–19], as well as to interpret

]
M a complex system and its global behavior. This problem electron transfer experiments [20] and time-resolved IR
has been formalized in the context of complex networks measurements [21, 22].
B
[2]asaquestionof“navigability”;i.e.,themechanismfor In this letter, the relation between the local proper-
. the efficient flow of information when the single network ties of the free-energy landscape and its global architec-
o
i nodes do not have a global view of the overall topology. ture is investigated by MD simulations of a 4 residues
b
As a matter of fact, many collective dynamical processes peptide, (GlySer) , and complex network analysis . In
- 2
q are driven by the presence of (usually hidden) local gra- particular, when the CSN of a fully-atomistic peptide is
[ dients [3]. reducedtothesubgraphcontainingonlyonelinkpermi-

1 The study of protein dynamics involve a similar prob- crostate pointing towards the most probable transition
v lem – the coupling between the fast atomic fluctuations, (i.e. following the transition gradient), the presence of
0 which are local, and the slow conformational changes, energy valleys and subvalleys and their equilibrium pop-
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which are global [4]. Those dynamical aspects have been ulations is naturally extracted as well as the hierarchy
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recently recognized to be crucial for protein function, of transitions between them. Hence, the fast local mo-
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. playing an important role in signalling, allosteric path- tionsbuildupthebackboneoftheglobalcommunication.
1
waysandenzymaticreactions[5,6]. Moleculardynamics The observed coupling between local and global dynam-
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(MD) simulations are playing an increasing role in com- ical properties is expected to occur in a large class of
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1 plementing the experimental results [7–9] which supply complex systems.
: useful , but limited, information to this question. The GlySerpeptideshavebeenusedforquitesometimeas
v
recent combination of computational and experimental flexible linkers (and are known to show poor secondary
i
X
studiesofaproteinenzymehaspointedoutthatthefast structure)forpolypeptidedynamics[23,24]. MDsimula-
r atomic fluctuations are partly correlated to the displace- tions using the Langevin algorithm [25] and the implicit
a
ments occuring in the catalytic reaction [10]. As yet, the solvation FACTS [26, 27] have been performed. A tra-
couplingbetweenthefastnanosecondtimescalesandthe jectory of 280 ns at 340 K was obtained and snapshots
functional relevant transitions occuring in the microsec- were saved every 140 steps for a total of 106 conforma-
ond to millisecond range largely remains obscure. tions. During the simulation the peptide visits several
Most descriptions of the free-energy surface governing different conformations characterized by an end-to-end
protein dynamics have been rather qualitative because distancebetween3.1and12.7˚A(Fig.1)indicatinglarge
of the lack of proper order parameters and the intrinsic structural fluctuations.
multidimensionality of the problem [11, 12]. These lim- The peptide microstates are defined as the inherent
itations have triggered the development of a completely structures (IS), i.e. the potential energy minima, of the
newarsenaloftoolsinspiredbynetworktheory[13]. The system [28, 29]. They are calculated minimizing all the
essential idea is to map the protein trajectory, obtained 106 snapshots along the trajectory, resulting in 3044 dif-
by computer simulations or experiments, on a confor- ferent IS. The IS are a natural, physically meaningfull,
mation space network (CSN), whose nodes represent the partition into microstates [30–32]. The conformation
2
14 5
12
4
10 ol]
m
[A]e 8 cal/ 3
e k
d 6 F [
∆ 2 V4
4 V1 V V3
2
2 1
0 0.25 0.5 0.75 1 0 0.2 0.4 0.6 0.8 1
Time [ns] Z
FIG. 1. Timeseries window of the end-to-end distance of the FIG. 2. (Color online) Cut-based free-energy profile for the
(GlySer) peptide. The peptide is extremely flexible, alter- (GlySer) peptide. Microstates assigned by the gradient-
2 2
nating between compact states (d ∼ 3.4 ˚A) and extended approach to the four most populated valleys are shown on
ee
conformations (d ∼10.8 ˚A). the profile in different colors. The CFEP represents the free-
ee
energysurfaceprojectedonthecumulativepartitionfunction
reaction coordinate Z [34], relative to a given reference mi-
space network (CSN) is built on top of this microstate crostate(inthiscasethemostpopulatedone). Intheinseta
profile is calculated on a subgraph made by the microstates
definition: the nodes and the links are the microstates
belongingtoV (triangles)andasmallergradient-cluster(cir-
(i.e. the IS) and their direct transitions observed during 4
cles).
the MD trajectory, respectively. The obtained network
is weighted and it is equivalent to a classical transition
matrix when the columns of the adjacency matrix (i.e. shown. The profile reveals that the peptide free-energy
the network links) are appropriately normalized to one. landscapeisrugged. Remarkably, theobtainedgradient-
Therelationbetweenfastlocalmodesandglobalchain clusters represent either a valley or a subvalley of the
rearrangements is investigated by constructing from the free-energylandscape(Fig.2). InTableIthepopulation
fullCSN,anewnetworkwithareducednumberoflinks. of the four most relevant free-energy basins detected by
For each network node, the transition with the highest the gradient-approach is compared against the popula-
probability (excluding self interactions) is kept, and all tions calculated by the minimum-cut method [16], which
others are deleted. These transitions define a gradient isoneofthemostaccurateapproachesforthistypeofcal-
in the network dynamics and result in a partitioning of culation[13]. Theresultsindicatethatthepopulationsof
the network into several disconnected minimum span- thegradient-clustersareaccurate,effectivelyreproducing
ning trees [33], called here gradient-clusters for conve- thecorrectthermodynamicsofthesystem. Thisispartic-
nience. (Gradient networks were originally introduced ularly relevant since the gradient-approach does not use
by Toroczkai and Bassler to study jamming [3], though anyglobalpropertyofthesystem,neitherintermsofbar-
in their case the gradient is defined on the nodes as a rier heights or microstates energies. On the other hand,
quenchedscalarfield). Followingthepathwaydefinedby the CFEP and the minimum-cut method do perform a
the most probable transitions leads to microstates lower global analysis of the network. These observations sug-
and lower in free energy, resulting in a kind of “steepest gest an interesting application of the proposed approach
descent pathway” on the free-energy surface. High en- toautomaticallydetectthepresenceofmetastablestates
ergymicrostates,intheneighborofafree-energybarrier, sampledbymultipleshortMDrunswiththeaimofbuild-
wouldconnecteithertoonevalleyoranother. Asamat- ing simplified Markov models [37].
teroffact,thenetworkcharacterizingthefree-energysur- The gradient-approach can be applied in an iterative,
faceissplitintoasetofdisconnectedminimum-spanning- heirarchical fashion, when the gradient-clusters are con-
trees representing the local attractors of the system dy- sidered themselves as nodes of a higher level CSN. In
namics. this case, the nodes and the links of the network are the
To better understand the nature of the gradient- gradient-clusters and the connections between them, re-
clusters (161 in total), a cut-based free-energy profile spectively. The iteration can be applied recursively until
(CFEP) [34] is calculated on the CSN and compared to all the microstates are merged into one cluster, which
theoutputofthegradient-partition. TheCFEPisbased is represented as a minimum-spanning-forest. The tree
on a network flux analysis following the idea that the structure obtained for the (GlySer) peptide is shown in
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network regions of minimum flow correspond to transi- Fig. 3. Link widths represent at which iteration step the
tion states [34–36]. In Fig. 2 the calculated CFEP is edgewasintroduced. Forexample, theV andV valleys
1 2
3
Valley Gradient Mincut <d > [˚A]
ee
population a population
V 0.429 0.428 4.167
1
V 0.120 0.120 4.253
2
V 0.068 0.064 10.960
3
V 0.037 0.051 4.160
4
a Totakeintoaccountoftheentropiceffects,gradient-clusters
separatedbyfree-energybarrierssmallerthankBT/10have
beenmerged.
TABLE I. Comparison of the populations of the found val-
leyscalculatedbythegradient-approachorbytheminimum-
cut method [16]. Populations are defined as the sum of the
number of occurencies calculated along the trajectory of the
microstatesassignedtoagivenvalley. Inthelastcolumnthe
averagevalueoftheend-to-enddistanced insideavalleyis
ee
given.
are merged at the first iteration, indicating that they in-
terconvertrapidly. Ontheotherhand,V ismergedtoV
4 1
at the fourth iteration indicating that this is the slowest
FIG. 3. (Color online) Gradient-minimum-spanning-forest of
transition to V . The gradient-minimum-spanning-forest
1 the (GlySer) peptide energy landscape. This free-energy
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represents, in an intrinsically multidimensional fashion,
representationconsistentlyrepresentsthefastrelaxationsbe-
the valleys of the free-energy landscape and their dy- tween the microstates inside a valley (star-like structures) as
namical organization in fast and slow relaxations. The wellasthedynamicalseparationbetweendifferentvalleys(ex-
iteration step at which two gradient-clusters are merged pressedbylinkthickness). Largerlinkwidthsindicateslower
together is a kinetic mesaure of the dynamical distance relaxations (see main text for details). The size of the nodes
is proportional to the microstate population (to avoid over-
between the two valleys.
crowiding only nodes with populations larger than 10−3 are
Concluding,wefoundthat,foranall-atompeptideMD
shown).
simulation, the fast local relaxations produce a partition
ofthe conformationalspaceintodisconnectedminimum-
spanning-trees. This partition reproduces the organiza-
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