The Static Modes - LAAS-CNRS

Eur. Phys. J. E 28, 17–25 (2009) DOI 10.1140/epje/i2008-10397-0

THE EUROPEAN PHYSICAL JOURNAL E

Regular Article

The Static Modes: An alternative approach for the treatment of macro- and bio-molecular induced-fit flexibility M. Brut1,a , A. Est`eve1 , G. Landa1 , G. Renvez1 , and M. Djafari Rouhani1,2 1 2

LAAS-CNRS, Universit´e de Toulouse, 7 avenue du Colonel Roche, F-31077 Toulouse, France Universit´e de Toulouse; UPS, France Received 31 July 2008 c EDP Sciences / Societ` Published online: 13 January 2009 –  a Italiana di Fisica / Springer-Verlag 2009 Abstract. We present a new competitive method for the atomic scale treatment of macromolecular flexibility called Static Mode method. This method is based on the “induced-fit” concept, i.e. it maps the intrinsic deformations of a macromolecule subject to diverse external excitations. The algorithm makes it possible to obtain a set of deformations, each one corresponding to a specific interaction on a specific molecular site, in terms of force constants contained in the energy model. In this frame, the docking problem can be expressed in terms of interaction sites between the two molecules, the molecular deformations being extracted from the pre-calculated Static Modes of each molecule. Some preliminary basic examples aimed at illustrating potential applications where macro- or bio-molecular flexibility is of key importance are given: flexibility inducing conformational changes in the case of furanose ring and flexibility for the characterization, including allostery, of poly(N-isopropylacrylamide)(P-NIPAM) active sites. We also discuss how this procedure allows “induced-fit” flexible molecular docking, beyond state-of-the-art semi-rigid methods. PACS. 87.15.A- Theory, modeling, and computer simulation – 33.15.Bh General molecular conformation and symmetry; stereochemistry – 87.15.-v Biomolecules: structure and physical properties

1 Introduction Progress in structural biology, in genome and proteome has led to target a vast number of proteins. But only a small fraction of them has been studied in detail, due to limitations of the most advanced characterization techniques, including modeling methodologies, and to the high complexity of their interactions. However, understanding and predicting the correlation between structure and function of biomolecules requires a concentration of the efforts and molecular modeling is intended to be a leading tool for the prediction of biological processes, as much for single molecules than for interactions [1]. Despite this fact, whereas changes of conformation are a key issue in biology, current algorithms are not adequate to explore the flexibility of macromolecules [2]. Introducing flexibility with an acceptable computational cost in already complex multi-level modeling strategies is a real challenge, and is essential to improve our understanding of both intrinsic properties of single molecule (folding) and interactions through the formation of macromolecular complexes (docking procedures). In order to provide blind predictions from native structures, many algorithms have already been developed and a

e-mail: [email protected]

proposed to the community [3]. However, the results remain costly in time and often unsatisfactory. This is particularly true when increasing the size of the biomolecules (number of atoms) and subsequently their natural propensity to undergo large amplitude conformational changes. For a long time, the “lock and key” theory, introduced by Fisher [4], has been used to explain docking processes. Along this line, rigid body-based models were used in calculations as soon as the computer capabilities became acceptable for this purpose. However, it is now well known that biological molecules are not rigid entities. On the contrary, their structure can be highly perturbed by external constraints. Koshland [5–7] was the first to underline the ability of an enzyme and its substrate to adapt to each other and further introduced the “induced-fit” theory. Jorgensen [8] extended this concept to the conformational changes undergone by two molecules during an interaction. From a modeling point of view, the ability of biomolecules to change their conformation can no longer be neglected in favor of “key-lock”-based models through interacting rigid or semi-rigid bodies. It is necessary to determine molecular flexibility, as well as to understand its role in the correlations between molecular structure and reactivity so as to predict correctly docking mechanisms.

18

The European Physical Journal E

Flexibility has therefore attracted much attention these last years and several methodologies have been proposed to include flexibility into existing modeling strategies. The main idea is to identify the most relevant flexible parts of biomolecules, for instance from calculated normal modes or data banks extracted from experimental observations, in order to obtain a semi-rigid model that is still manageable by current modeling tools. This evolution towards “adaptable objects” through flexibility does not really satisfy the “induced-fit” concept. To decrease the computational cost, while offering an alternative way of introducing flexibility through the concept of “induced fit”, we have developed a new computing method: the Static Mode method. This method results in the mapping of all possible deformations of a molecule subject to various external excitations. A given mode corresponds to an excitation on a specific atom in a specific direction. In this scheme, the concept of normal modes [9], which are natural vibrations of a molecule, is abandoned in favor of the permanent deformations caused by interactions with other molecules. In Section 2, we will present the origin of the concept and the algorithm for calculating Static Modes, before comparing this approach to Molecular Dynamics (MD) and Normal Mode Analysis (NMA), which are the most common methods used to describe macromolecular interactions. Results on two particular examples are presented and discussed in Section 3.

2 Methods 2.1 Origin of the concept The fundamental idea of Static Modes is to simplify a MD-based treatment of the docking. Given the computation cost related to MD, this simplification is necessary if one intends to investigate a significant number of initial situations to end up with a scoring procedure. In this scope of simplification, Static Modes can be considered as an alternative to NMA, although with completely different basic principles. Indeed, in NMA, single modes with the lowest frequencies are considered to be responsible for permanent molecular deformations allowing the docking. However, at room temperature, only normal modes with vibration frequencies much higher than 1013 Hz, can still be considered as well-defined modes. All other modes are jammed together. It is obvious that such high frequencies are not compatible with reaction times occurring in biological processes. In order to overcome this difficulty, we propose to replace thermally activated free vibrations of the molecules by permanent deformations induced by external interactions. Comparing these two approaches with MD, NMA considers only the intramolecular dynamics while Static Modes are merely oriented towards the intermolecular dynamics, i.e. interaction dynamics. It is obvious that our scheme is closer to actual docking processes, where, after a series of transient oscillations, the molecular vibrations

settle to a new set of normal modes, around the modified geometry of molecules in interaction. It is worth mentioning that the transient oscillations, during the docking process, cannot be easily related to the Normal Modes of the single molecule, but can only be reached through MD treatment. To save computing time with respect to MD treatment, we suggest to shortcut the transient-state vibrations and calculate directly the steady-state deformations. This suggestion assumes that biological processes are slower than the transient-state duration. Transientstate damping can be roughly estimated using viscous forces exerted on moving bodies in aqueous media: the classical Stokes formula gives 10−12 s.

2.2 The algorithm As in NMA, the Static Modes are based on the Hessian matrix which contains the second derivatives of the total energy with respect to atomic coordinates. We will discuss in Section 2.3 the applicability conditions of Static Modes with respect to MD and NMA. The Hessian matrix elements are functions of the force parameters used in classical semi-empirical interatomic potentials. Using quantum ab initio calculations, the matrix elements can be used to determine these force field parameters. In any case, the Static Modes can be obtained from any type of energy model: quantum or classical, sophisticated or simplified ones. The choice of the energy model depends on the size of the system and the required accuracy. Discarding the translation and rotation coordinates, one ends up with a (3N − 6) × (3N − 6) matrix, N being the number of atoms in the molecule. The next step is to impose a constraint, in the sense of Lagrange constraints, on the molecule. The total energy can be expanded into a power series of the atomic coordinates as E = E0 +Hi,j xi xj + Ki,j,k xi xj xk + Li,j,k,l xi xj xk xl + . . . , (1) where Hij are the Hessian matrix elements, K, L, . . . the matrices representing the successive derivatives of the energy, and xi the Cartesian coordinates of the atomic displacements from a given equilibrium configuration. Expressing the force vector as the gradient of the energy, written in terms of the Hessian matrix, one obtains Fi = Σj H  ij xj .

(2)

Using the harmonic approximation, i.e. neglecting all high-order energy derivatives, H  is just the Hessian matrix. The effect of anharmonic terms is discussed in Section 2.3. At equilibrium and in the absence of external forces, all xj = 0. Applying a force on a single atom in a given direction, i.e. all Fi except one vanish, one can find the atomic displacements as the solution of the set of equations (2). Alternatively, one can assume a given displacement xl , resulting from the application of the force Fl . Both approaches are equivalent: in the former, the displacements

M. Brut et al.: The Static Modes: A new approach to treat macromolecular flexibility

are normalized to the applied force while in the latter, they are normalized to the displacement xl . We adopt this second approach where the fixed xl constitutes the Lagrange constraint while Fl acts as a reaction force whose knowledge is not necessary in the Lagrangian formalism. Although this second approach is more formal and complex, we think that it can be more easily transposed to internal coordinates which can then be considered as generalized coordinates, in the Lagrangian sense. Moreover, dealing with repulsive forces between closely spaced atoms, this second approach allows the use of the hard-sphere approximation. This approximation is very useful since these repulsive forces are difficult to estimate: they present large variations over small atomic displacements and their exact knowledge is not required. From a practical point of view, the equation containing Fl , i.e. the l-th line of the Hessian matrix, is now a constraint equation and should be eliminated from the set of (2). At the same time, all the terms containing xl , i.e. the l-th column of the Hessian matrix, are no more unknown. Denoting this column by B and the remaining (3N − 7) × (3N − 7) matrix by A, we have to solve the set of equations AX + B = 0, where X represents the atomic displacements imposed by the constraint. The solution of the above set of equations can be obtained by various conventional algorithms. Gauss-Seidel method is well-adapted to molecular deformations, because it allows the smooth propagation of the deformation from the excitation source, and has been adopted in this work. This solution shows the molecular deformation corresponding to one specific constraint and is called a Static Mode. Once a Static Mode, normalized to the displacement xl has been determined, the l-th equation in the set of (2) can be used to calculate the reaction force Fl . If desired, the mode can then be easily normalized to this force. Also, the energy associated to the mode can be simply determined. Given a biomolecule, all Static Modes, or few specific ones, can be determined by suppressing sequentially one of the Hessian matrix lines and solving the remaining set of (3N −7)×(3N −7) equations. Since these modes are characteristics of a single molecule, they can be stored in a data bank for further use, molecule docking for example, simplifying the procedure and saving computer time. In the case where some molecular sites can definitely be discarded as interaction sites, the corresponding Static Modes need not be calculated. The computing time is then reduced by the same amount. 2.3 Discussion MD is usually considered as the most straightforward method to investigate the evolution of systems in interaction, since it applies Newton law, taking carefully all forces into account. However, when applied to complex systems, such as biomolecules, the prohibitive computing time is not the only difficulty encountered. On the one hand, knowledge of all details of movement makes their interpretation so difficult that sophisticated statistical treatments should be applied to extract the basic

19

information needed, out of a huge data pile. Even though, the information is usually subject to large fluctuations. These fluctuations do not only result from the statistical treatments, but they also reflect the oscillatory nature of the complex system in real experimental conditions, in particular the ambient temperature. On the other hand, the phase space associated to these systems is so large that it can never be covered by a single experiment, whatever its duration. To get real insight into this phase space, one should theoretically perform a large number of experiments, with different initial conditions, before applying the statistical treatment. This is obviously not possible in practice. Therefore, results obtained with a single or a few experiments lack also confidence, in addition to large fluctuations. These difficulties reflect the fact that the use of a perfect model is probably not the best approach to catch the basic properties of a complex system. Static Modes constitute one simplified approach which allows reaching the final deformations by skipping all transient oscillations which are damped before the intermolecular interaction actually occurs. NMA is another simplified approach intended to reproduce the instantaneous picture of these oscillations. However, its applicability conditions limit severely its use in many practical situations. The first condition is that NMA can only be performed around an equilibrium position. This means that NMA cannot be applied to two molecules in relative motion. Moreover, in complex systems, the configuration corresponding to the minimum energy should be determined very carefully. Otherwise, negative eigenvalues are found for the Hessian matrix, giving rise to imaginary frequencies. This accurate minimization procedure is itself computer time consuming. The equilibrium condition is not required in the Static Mode approach, since these modes are excited by the accumulation and/or variation of external forces which occur as a result of the relative motion between two molecules. Also, accurate determination of the configurations is never needed since the configurations are in permanent evolution. The second condition is related to the harmonic nature of the applied forces. As pointed in Section 2.2, H  is no more the Hessian matrix when taking anharmonic terms into account. Rather, H  ij will depend on atomic coordinates. The set of equations (2) is still valid but is no more a linear set. Its solution can be easily found using the previous algorithm, provided a small increase of the computing time is allowed. Considering NMA, the use of coordinatedependent matrices is completely forbidden, since the harmonic approximation is the prerequisite for their determination. Although non-harmonic terms are at the origin of multiple configurations observed in the vast majority of biomolecules, they have not been taken into account in the preliminary illustrations treated below, where we do not look configuration changes. Such examples are presently under investigation. The computing cost of the Static Mode determination is very low, as can be seen through the following examples. The basic systems presented here, consisting of less than

20

The European Physical Journal E

one hundred atoms, require less than 1 CPU second (Intel Xeon dual core, 1.6 GHz). More complex molecules, as proteins for example, use few hours: we have treated the case of HIV-1 protease, consisting of 3128 atoms, in 7:20 CPU hours. We should mention that, once calculated, the Static Modes can be stored and reused in further applications. In terms of algorithm complexity, the calculation of all Static Modes is equivalent to a matrix inversion, which is much easier than the matrix diagonalization, necessary for the determination of the normal modes. The difficulty to diagonalize large matrices, particularly the convergence problems encountered, has often led to the reduction of biomolecules to their backbones and to an oversimplification of the interaction potentials [9,10]. Simplified potentials with an appropriate choice of the parameters can sometimes reproduce experimental observations, but the analytical treatment remains far from basic physical interactions. The complete calculation of vibrational spectra of large biomolecules has not been frequently performed. Finally, the fact that inter-atomic interactions can be considered as negligible beyond a cut-off radius, simplifies greatly the solution of equations leading to Static Modes, while it has no effect on the diagonalization procedure.

3 Results and discussion The purpose of the following subsections is to provide some preliminary illustrations of the Static Modes capabilities through basic examples. Actually, beyond the technical aspects including the expected gain in CPU time, the method pictures macromolecular flexibility in an original way, that is described here after. We already pointed that the method can be used whatever the energy model. All reported results are based on classical potentials or quantum cluster models, via AMBER8 [11] or GAUSSIAN03 [12] packages, respectively. For each investigated example, all the Static Modes of the molecule are calculated within the harmonic approximation by solving the linear set of equations. The Static Modes and related post treatment data are emanating from our code FLEXIBLE. The post treatment procedures concern the use of the raw Static Modes calculated once, for an application to a specific problem. In the present work, the post treatment of data has two objectives; the first is to find the direction of the force to be applied on each atom, in order to optimize a given physico-chemical property of the molecule, mainly the opening of hydrogen type bonds or the modification of its characteristic angles. The second objective is to compare the optimized forces on various atoms to decide, as a guide to experimentalists, which action could be taken to enhance or inhibit an active site. While the first objective is well formalized and can be implemented by an algorithm, the second will need more expertise. Turning now to the first objective, we should remember that Cartesian coordinates are useful to define the vector components of the forces, but they are no more appropriate to characterize physico-chemical properties of a given molecule, since they depend tightly on the arbitrary choice of the reference frame. For this purpose, the rele-

vant parameter is the direction of the applied force optimizing the desired deformation, which explicitly refers to the molecule orientation. We are faced to a constrained optimization problem, where the force magnitude should be conserved when changing its orientation. Use of Lagrange multipliers is a classical solution to this problem which, however, remains three dimensional, since it only involves the three components of the force vector applied on a single atom. In the simplest cases treated in this article, where we use the harmonic approximation for the Hessian matrix, the Lagrange multiplier solution reduces to a simple diagonalization of a 3×3 matrix. In the following examples, we are searching for maximum sugar phase changes or maximum interatomic distances. In these cases, the optimized direction is given by the eigenvector corresponding to the largest eigenvalue of the matrix. In the following, we address three particular preliminary examples: conformational changes of a single molecule, exploration of an active site, and perspectives on docking procedures based on Static Modes, to illustrate our approach. 3.1 Molecular conformational changes Investigating conformational changes is of major importance to understand the function of a molecule and its ability to interact with another molecule or its environment. We have chosen the case of the furanose ring to illustrate this aspect. In DNA, furanose ring is known to be highly flexible [13], its conformational changes are highly investigated, experimentally [14] as well as by calculation [15] or modeling [16]. It behaves like a pivot between the backbone and the rigid bases, and so, plays an essential role in the global conformational changes in DNA structure. 3.1.1 Pseudorotation concept As the furanose ring is never planar, it can adopt multiple conformations. These conformations, called puckering modes, can be described by the pseudorotation concept, introduced by Sundaralingam et al. [17,18], who established relationships between the five endocyclic torsion angles ν0 to ν4 : ν0 : C4 -O4 -C1 -C2 ; ν1 : O4 -C1 -C2 -C3 ; ν2 : C1 -C2 -C3 -C4 ; ν3 : C2 -C3 -C4 -O4 ; ν4 : C3 -C4 -O4 -C1 (see Fig. 1). Puckering modes are thus uniquely defined by two angles: the phase angle of pseudorotation P and the maximum out-of-plane pucker νmax , described by equations (3) and (4), respectively. (ν4 + ν1 ) − (ν3 + ν0 ) , (3) 2ν2 (sin 36◦ + sin 72◦ )  ν   2  νmax =  (4) . cos P Overall puckering modes can be represented on a pseudorotation cycle by plotting P and νmax on a polar graph. In a single strand, furanose ring can be found in different conformations, but, as shown by experimental structures, the main ones are typically C3 -endo (called north tan P =

M. Brut et al.: The Static Modes: A new approach to treat macromolecular flexibility

21

by spectroscopists because P is in the range −1◦ ≤ P ≤ 34◦ ) and C2 -endo (called south, in the range 137◦ ≤ P ≤ 194◦ ). In a duplex, sugar is usually fixed in north conformation in A-form duplex and south in B-form duplex [15].

to undergo phase changes. Notably, focusing on Figure 2, we can distinguish particular transition pathways. This means that, for a given transition or a particular puckering modes, it is possible to determine what constraints should be applied and to choose the more relevant one. For example, for a south/ north transition, the more relevant force is the one applied to O3 atom, with a direction represented on Figure 4. Also, the largest phase change on Figure 2 corresponds to a constraint imposed on the carbon atom of thymine methyl group. As shown on Figure 4, it is interesting to note that this carbon atom is situated far from the sugar. In this case, the conformation obtained is represented on Figure 5. This figure shows the maximum phase change state appearing on the pseudorotation cycle (2 E) compared to the phase change of the sugar in its equilibrium state (21 T). In this example, a complete chemical pathway could be followed step by step, by applying adequate deformations derived from the Static Modes.

3.1.2 Results

3.2 Active site and allostery

We consider a desoxy-thymidine-monophosphate (dTMP), studied in a single form and in a duplex with its complement, a desoxy-adenosine-monophosphate (dAMP). The molecular structures presented here result from a classical energy minimization made with AMBER8 suite of programs [11] in implicit solvent using the ff94 force field. From these structures, we have calculated the Static Modes for the whole molecules and determined, for each atom, along the procedure described above, the direction of the force which maximizes the phase changes. Figures 2 and 3 show the conformational changes of dTMP sugar in a single and in a double DNA strand, respectively. In the equilibrium state (red points), both sugars are in south conformation: P = 150.86◦ and νmax = 35.29◦ for the single strand sugar, P = 151.52◦ and νmax = 35.21◦ for the double-strand sugar. These values are within the experimental range [19,20]. Each point on the graph represents a possible change of conformation of the molecule after application, on a given atom, of a normalized force in an optimized direction. From Figures 2 and 3, we observe a slight difference of the cloud distribution around the equilibrium. The cloud in the single-strand nucleotide is spread towards few specific directions, including a south/north direction, while the double strand exhibits a more compact and anisotropic behavior, perpendicular to the southnorth direction. Therefore, it can be concluded that the sugar of the single-strand nucleotide is able to change its conformation more easily, at least along these specific axes. On the contrary, the sugar of the same nucleotide included in a double strand is much more rigid and does not exhibit south/north direction to change its conformation. In addition to these direct preliminary conclusions, the study of this type of graphs allows deep and simple exploration of the sugar flexibility in relation with the global structuring of the DNA. A direct observation of the graph point distribution can indicate the propensity of the sugar

Static Modes constitute a suitable tool to study a known molecular active site or even to search for new ones. As an example, we treat here the case of the poly(N-isopropylacrylamide)(P-NIPAM), a thermosensitive polymer which is hydrophilic below its low critical solution temperature (LCST) and hydrophobic above this temperature [21,22]. In its hydrophilic phase, PNIPAM is observed in a coil form, swelled by water molecules bonded to two adjacent monomers by hydrogen bonds. Figure 6 shows two monomers (a) and (b), connected by a hydrogen bond, in the absence of water. Normally, these hydrogen bonds break before incorporation of water molecules into the active sites. We can assume that the swelling of the polymer starts by the opening of these hydrogen bonds before they break completely. The model used hereafter is composed of two monomers. Its equilibrium structure is found by minimizing its total energy, calculated within the frame of the density functional theory and using the quantum-chemical software GAUSSIAN03 [12]. Calculations use the B3LYP functional [23] and a 6-31++G** basis as described in more details in [24]. Using this equilibrium structure, Static Modes are calculated and the force directions on each atom are optimized to get the maximum hydrogen bond opening. The two-monomer PNIPAM is structurally divided into three parts, as shown in Figure 6: (a) and (b) are the monomers that share the active site into their polar regions, (bb) is the backbone connecting the two monomers. We have represented on Figures 7, 8 and 9 the opening of the hydrogen bonds as a function of the distance between the constrained atoms and the active site center, for each one of the three parts of our model (a), (b) and (bb), respectively. Comparing Figures 7 and 8, it appears that the molecule flexibility response is different when forces are applied on (a) or (b) parts of the molecule. This can be understood because of the chemical asymmetry of the

Fig. 1. Furanose ring: scheme of atomic numbers and endocyclic torsion angles.

22

The European Physical Journal E

Fig. 2. (Colour online) Pseudorotation cycle of the furanose ring of a dTMP. The red point represents the equilibrium state, black points are the possible conformational changes predicted by the Static Modes.

Fig. 3. (Colour online) Pseudorotation cycle of the furanose ring of a dTMP bonded to a dAMP. The red point represents the equilibrium state, black points are the possible conformational changes predicted by the Static Modes. The furanose ring seems to be more rigid in this state than in a single state.

active site (C=O . . . H–N). We also observe, by examining the maximum amplitudes in Figures 7 and 8, that the monomer (a) has little influence in the hydrogen bond opening, compared to monomer (b). Furthermore, both monomers exhibit allostery, as the more favorable constraints are not the ones applied on the close neighbors of the active site, neither on the active site itself. Another important point is that constraints applied on different atoms, at the same distance from the active site, do not induce necessarily the same opening response and do not show similar effects in the hydrolysis process. This is a clear indication that some local-

ized modes, for example carbon atoms of the isopropyl hydrophobic monomer (b) termination, are crucial to allow for the hydrolysis of the polymer. In other words, a constraint imposed on these terminations could enhance or hinder the natural propensity of the polymer to undergo its phase transition. Finally, we observe that the constraints imposed on the backbone have no influence on the active sites, compared to the constraints imposed on the monomers (Fig. 9). These conclusions are now shown in a different way on Figure 10, which represents the forces applied on each atom to obtain a maximum opening. The force vectors are

M. Brut et al.: The Static Modes: A new approach to treat macromolecular flexibility

Fig. 4. (Colour online) Representation of the forces applied to obtain an optimized phase change (on the carbon atom of the thymine methyl group) and an optimized south/north transition (on the O3 atom). The norme represented is proportional to the hydrogen bond opening.

23

Fig. 7. Hydrogen bond opening resulting from the excitation of the atoms situated in the monomer (a) normalized to the excitation force (in ˚ A).

Fig. 5. (Colour online) Two possible puckerings of the furanose ring of the single dTMP: equilibrium state of our structure (21 T) on the left and the maximum phase change visible on Figure 2 (2 E).

Fig. 8. Hydrogen bond opening resulting from the excitation of the atoms situated in the monomer (b) normalized to the excitation force (in ˚ A).

Fig. 6. (Colour online) Atomic representation of a PNIPAM dimer. Both monomers (a) and (b), and the backbone (bb) are annotated.

now proportional to the opening amplitude, so that the longer the arrow is, the easier it is to break the hydrogen bond and to open the active site. This type of representation is interesting to map quickly the atoms of a molecule where external excitations have the most important effect for a desired action. It revels here a particular area on the strand (b) which has a real influence on the active site and can be studied in detail for a possible experimental approach. In the present case, it reveals a particular and unexpected area on the monomer (b) which has a real influence on the active site and can be studied in more details for the setting-up of an experimental investigation.

Such a study of an active site could be of major importance to find new mechanisms for activation or inhibition: these mechanisms, in this view, would be to design experiments where the flexibility would be altered in a predefined way. We point out the fact that the method can also be systematically used to screen potential allosteric sites. Then, it also would allow an understanding of the complex relation of these interconnected sites. 3.3 General discussion on the docking procedure via Static Modes Up to now, we have illustrated how the Static Modes can be used on a single molecular entity. The exploration of conformational changes is straightforward as the deformations derived from the Static Modes may help to define specific directions of the potential hypersurface. Considering now the problem of docking, a specific docking-based Static Mode algorithm should proceed through identified

24

The European Physical Journal E

Fig. 9. Hydrogen bond opening resulting from the excitation of the atoms situated in the backbone (bb) normalized to the excitation forces (in ˚ A).

interacting sites between the two molecules. The constraints imposed on these sites create a strain within each of the two molecules, according to Static Modes calculated previously and stored in a data bank. The molecular deformations are straightforward and driven by the evolution of intermolecular forces exerted on the interaction sites, as the two molecules move in space. The dynamic variables during the docking process are therefore restricted to the coordinates of atoms in direct interaction. The dynamics of the remaining atoms is already contained in the Static Modes. The number of atoms in direct interaction being usually much smaller than the total number of atoms within the biomolecules, the docking procedure can be performed in a very short time. This allows the investigation of a large number of situations where different sets of interaction sites can be tested.

Fig. 10. (Colour online) Representation of the forces applied to obtain an optimized active site opening. The norme represented is proportional to the hydrogen bond opening.

conformational changes of a molecule can be described in terms of specific deformation modes. Along the same line, we show how an active site can be characterized via the identification of the most relevant deformation inducing activation/desactivation of this site. In this frame, the degree of allostery of the molecule can be easily reached. The method thus offers an opportunity to determine new targets for drug design. Finally, we indicate how the Static Modes approach can be used to perform flexible docking. We believe that further development of this approach and its integration in multi-level modeling strategies will be a step forward into the understanding of macromolecular flexibility and its role in the physico-chemistry of biomolecular interactions. We thank CALMIP supercomputer center for CPU resources. We also thank projects ANR-NANOBIOMOD and ITAVALMA for financial support.

4 Conclusion In conclusion, we have developed and presented an original “induced-fit” approach aimed at evaluating macromolecular flexibility. This new method, namely the Static Modes method, allows the systematic determination of the deformation field of a molecule submitted to an external and localized excitation. Thus, each mode corresponds to a specific interaction on a specific molecular site, in terms of force constants derived from a given energy model. Our approach is expected to be highly competitive: i) on the conceptual side, full flexibility is here considered in close relation to potential interactions, anharmonicity can be easily introduced, ii) on the computational side, the Static Modes calculated once and stored in a data bank, can be re-used to investigate several sets of active sites or in a great number of applications where macromolecular flexibility is an issue, saving computing time. As preliminary illustrations, we have calculated the whole sets of Static Modes related to some simple molecules and mapped their flexibility as a function of various external excitations. More specifically we explain how

References 1. H.J. Wolfson, M. Shatsky, D. Schneidman-Duhovny, O. Dror, A. Shulman-Peleg, B. Ma, R. Nussinov, Curr. Protein Pept. Sci. 6, 171 (2005). 2. A.M.J.J. Bonvin, Curr. Opin. Struct. Biol. 16, 194 (2006). 3. N. Brooijmans, I.D. Kuntz, Annu. Rev. Biophys. Biomol. Struct. 32, 335 (2003). 4. E. Fischer, Ber. Deutsch. Chem. Ges. 27, 2985 (1894). 5. D. Koshland, Proc. Natl. Acad. Sci. U.S.A. 44, 98 (1958). 6. D. Koshland, Science 142, 1533 (1963). 7. D. Koshland, Angew. Chem. Int. Ed. Engl. 33, 2375 (1994). 8. W. Jorgensen, Science 254, 954 (1991). 9. F. Tama, Y.-H. Sanejouand, Protein Eng. 14, 14, 1 (2001). 10. K. Hinsen, Proteins: Struct. Funct. Genet. 33, 417 (1998). 11. D.A. Case, T.A. Darden, I.T.E. Cheatham, C.L. Simmerling, J. Wang, R.E. Duke, R. Luo, K.M. Merz, B. Wang, D.A. Pearlman, M. Crowley, S. Brozell, V. Tsui, H. Gohlke, J. Mongan, V. Hornak, G. Cui, P. Beroza, C. Schafmeister, J.W. Caldwell, W.S. Ross, P.A. Kollman, AMBER8, University of California, San Francisco (2004).

M. Brut et al.: The Static Modes: A new approach to treat macromolecular flexibility 12. M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, J.A. Pople, GAUSSIAN03 ReV C.01, Gaussian, Inc.: Pittsburgh (2003). 13. M. Levitt, A. Warshel, J. Am. Chem. Soc. 100, 2607 (1978).

25

14. V.N. Bartenev, N.G. Kamenevaand, A.A. Lipanov, Acta Crystallogr. B 43, 275 (1987). 15. K. Arora, T. Schlick, Chem. Phys. Lett. 378, 1 (2003). 16. G.A. Meints, T. Karlsson, G.P. Drobny, J. Am. Chem. Soc. 123, 10030 (2001). 17. C. Altona, M. Sundaralingam, J. Am. Chem. Soc. 94, 8205 (1972). 18. S. Rao, E. Westhof, M. Sundaralingam, Acta Crystallogr. A 37, 421 (1981). 19. W. Saenger, Principles of Nucleic Acid Structure (Springer-Verlag, 1984). 20. H.P.M. de Leeuw, C.A.G. Haasnoot, C. Altona, Isr. J. Chem. 20, 108 (1980). 21. M. Heskins, J.E. Guillet, J. Macromol. Sci. Chem. A 2, 1441 (1968). 22. H.G. Schild, Prog. Polym. Sci. 17, 163 (1992). 23. A.D. Becke, J. Chem. Phys. 98, 5648 (1993). 24. A. Est`eve, A. Bail, G. Landa, A. Dkhissi, M. Brut, M. Djafari Rouhani, J. Sudor, A.M. Gue, Chem. Phys. 340, 12 (2007).