Forces and Energetics of Hapten-Antibody Dissociation: A ... · Forces and Energetics of Hapten-Antibody Dissociation: A Biased Molecular Dynamics Simulation Study Emanuele Paci1,2,

doi:10.1006/jmbi.2001.5103 available online at http://www.idealibrary.com on J. Mol. Biol. (2001) 314, 589±605

Forces and Energetics of Hapten-AntibodyDissociation: A Biased Molecular DynamicsSimulation Study

Emanuele Paci1,2, Amedeo Caflisch3, Andreas PluÈ ckthun3

and Martin Karplus1,4*

1Laboratoire de ChimieBiophysique Institut Le BelUniversiteÂ Louis Pasteur4 rue Blaise Pascal67000, Strasbourg, France2Oxford Centre for MolecularSciences New ChemistryLaboratory, University ofOxford, South Parks RoadOxford OX1 3QT, UK3Biochemisches InstitutUniversitaÈt ZuÈ rich,Winterthurerstrasse 190CH-8057 ZuÈ rich, Switzerland4Department of Chemistry andChemical Biology, HarvardUniversity, 12 Oxford StreetCambridge, MA 02138, USA

E-mail address of the [email protected]

Abbreviations used: AFM, atomicBDM, biased molecular dynamics; Ccomplementarity-determining regioforce molecular dynamics.

0022-2836/01/030589±17 $35.00/0

The unbinding of ¯uorescein from the single-chain Fv fragment of the4D5Flu antibody is investigated by biased molecular dynamics with animplicit solvation model. To obtain statistically meaningful results, alarge number of unbinding trajectories are calculated; they involve a totalsimulation time of more than 200 ns. Simulations are carried out with atime-dependent perturbation and in the presence of a constant force. Thetwo techniques, which provide complementary information, induceunbinding by favoring an increase in the distance between the ligandand the antibody. This distance is an appropriate progress variable forthe dissociation reaction and permits direct comparison of the unbindingforces in the simulations with data from atomic force microscopy (AFM).The time-dependent perturbation generates unfolding pathways that areclose to equilibrium and can be used to reconstruct the mean force; i.e.the derivative of the potential of mean force, along the reaction coordi-nate. This is supported by an analysis of the overall unbinding pro®leand the magnitude of the mean force, which are similar to those of theunbinding force (i.e. the external force due to the time-dependent pertur-bation) averaged over several unbinding events.

The multiple simulations show that unbinding proceeds along a ratherwell-de®ned pathway for a broad range of effective pulling speeds.Initially, there is a distortion of the protein localized in the C-terminalregion followed by the ¯uorescein exit from the binding site. This occursin steps that involve breaking of speci®c electrostatic and van der Waalsinteractions. It appears that the simulations do not explore the same bar-riers as those measured in the AFM experiments because of the muchhigher unfolding speed in the former. The dependence of the force on thelogarithm of the loading rate is linear and the slope is higher than in theAFM, in agreement with experiment in other systems, where differentslopes were observed for different regimes. Based on the unbindingevents, mutations in the 4D5Flu antigen binding site are predicted toresult in signi®cant changes in the unbinding force.

# 2001 Academic Press

Keywords: single-chain antibody; ¯uorescein; binding; moleculardynamics; atomic force microscopy
*Corresponding author
ing author:

force microscopy;DR,

n; CFMD, constant

Introduction

Understanding the binding of a ligand to itsmacromolecular receptor is important both for itsfundamental interest and as an aid in designingnew drug candidates. Recently data from equili-brium and kinetic measurements have beensupplemented by single molecule experiments inwhich an external force is applied to dissociate theligand-receptor complex.1 ± 11 Molecular dynamics

# 2001 Academic Press

Figure 1. Model14 of the scFv (ribbons) bound to ¯u-orescein (thick line). (a) The vector rFA joining C-2 of ¯u-orescein to the C terminus of the antibody, whichde®nes the reaction coordinate, is in the plane of theFigure. (b) View of the scFv fragment from the haptenbinding site. (c) Fluorescein structure with atom typesand atomic partial charges; H (hydrogen), O (carbonyloxygen), OS (ester oxygen), OH1 (hydroxy oxygen), OC(carboxy oxygen), C (carbonyl carbon); all the others areCR1E (extended carbon in aromatic ring with onehydrogen atom). This Figure was made with the pro-gram MOLMOL.53

590 Simulations of Hapten-Antibody Dissociation

simulations12 which complemented the originalatomic force microscopic analysis of the streptavi-din-biotin complex1,2 gave some insights into theinteractions along the dissociation path.

One binding phenomenon of particular interestconcerns the interactions between an antibody andits strongly binding hapten. PluÈ ckthun and co-workers6,13 have recently used atomic forcemicroscopy (AFM) to measure the binding forcebetween recombinant antibody single-chain Fv(scFv) fragment14 and ¯uorescein and compared itto the results measured in solution binding; anscFv fragment is the minimal part of an antibodymolecule that includes the complete antigen-bind-ing site.15 ± 17 Fluorescein is ideal for such a studybecause it is a rigid molecule, which avoids com-plications arising from more than one confor-mation of the ligand.

Here, we report the results of multiple moleculardynamics simulations of the unbinding process.We use a molecular-mechanics potential18 in theCHARMM program,43 an implicit solvent model,EEF119 and a time-dependent or constant force per-turbation applied to a suitable reaction coordinate.In analogy with the AFM experiments, the reactioncoordinate was de®ned as a function of the dis-tance between the C terminus of the scFv fragmentand the C2 atom of ¯uorescein (see Figure 1(a)).The methods employed are the same as those usedto induce unfolding of b-sandwich and a-helicalproteins on a computationally accessible time-scale.20,21

Computational limitations require that the simu-lations separating the antigen from the antibody beperformed about seven to nine orders of magni-tude faster than the AFM experiments; e.g. in 1 nsinstead of 10 ms to one second. The biased molecu-lar dynamics (BMD) method is well suited for sucha study because the time-dependent perturbationpromotes unbinding by adding a minimal pertur-bation that has little effect on the short-timedynamics of the molecule and allows a uniformsampling of the whole range of values of the reac-tion coordinate. The application of a constant force,while more easily understandable in experimentalterms, generally requires a stronger perturbationand induces a stepwise unbinding, so that mostlymetastable states (i.e. states that are stable on thetimescale of the simulation) are sampled at differ-ent force magnitudes. The use of an implicit sol-vent model means that the solvent respondsadiabatically to a change of the reaction coordinate,as it would on the millisecond timescale. Becausethe implicit solvent simulations are relatively fast,multiple simulations can be performed to obtainstatistically meaningful results.

The present simulation methodology differssomewhat from the original study of the biotin-streptavidin unbinding12 and subsequent work onthe same system,22,23 as well as that on anotherantibody-hapten complex.24 Either no solvent orexplicit solvent were used in these studies. In theunbinding simulations, one atom of the receptor

was kept ®xed and one atom of the ligand wasattached to a spring that was pulled at constantvelocity, instead of the variable or constant biasingforce. As in the present work, the simulation time

Simulations of Hapten-Antibody Dissociation 591

was many orders of magnitude shorter than thetimescale of the experiment.

The present simulations examine the response ofboth the ligand and the receptor to different exter-nal perturbations that have the same macroscopiceffect, that of inducing unbinding over a broadrange of timescales. The main goal is to determinethe magnitudes of the forces and the energeticsinvolved, and to probe the atomic interactions thatcontribute along the unbinding pathway(s). By car-rying out multiple simulations (77 out of 85 resultin full unbinding) we are able to show that theaverage unbinding pathway and the force alongthat pathway have rather well-de®ned averageproperties. Moreover, the mean force (i.e. thederivative of the potential of mean force) along theaverage unbinding pathway is very similar to theaverage external force obtained from the time-dependent biasing perturbation. This indicates thatthe perturbation is such that unbinding pathwaysare sampled under quasi-equilibrium conditions,which makes it possible to estimate free energypro®les. Thus, the present equilibrium approach,which corresponds to the so-called blue moonmethod,25 is complementary to the non-equilibriumfree energy simulations proposed recently.26,27

The energetics during unbinding are found to bedominated by the intermolecular energy with thevan der Waals and electrostatic contribution chan-ging in a corresponding fashion. Further, there is apartial compensation between total electrostaticand solvation energy, as expected, and the vari-ation of the intra-receptor energy is small. Theenergetics and mechanism described here for thedissociation of ¯uorescein from an antibody scFv

fragment provide an example of mechanicallyinduced hapten-antibody dissociation.

Results

The control simulation

The scFv-¯uorescein complex is stable over the2.4 ns time of the control run (Figure 2). The over-all conformation and secondary structural elementsare preserved, as is the hapten-binding site. Asexpected, the largest ¯uctuations (not shown) arelocalized in the linker atoms (see Materials andMethods). This is consistent with an analysis of the15N NOE data for the complex between the scFvfragment of the antibody McPC603 and phos-phorylcholine.28 During the last 800 ps of the con-trol run, the average non-linker RMSD from theoriginal model structure (with superposition of thenon-linker Ca atoms) is 2.7 AÊ for Ca atoms and3.3 AÊ for all heavy atoms. The RMSD averagedover the ®nal 800 ps for the Ca atoms in the frame-work region is 2.2 AÊ (maximum of 2.9 AÊ ) and forthe Ca atoms in the complementarity-determiningregion (CDR) loops (residues 24-39, 55-61, 94-102,175-179, 194-212 and 245-251) is 3.3 AÊ (maximumof 4 AÊ ). The CDRs of the VL domain deviate by3.2 AÊ on average, while the CDRs of the VH

domain deviate by only 1.6 AÊ . These values arecomparable to the 3.2 AÊ deviation from the X-raystructure for the backbone heavy atoms reportedby Heymann & GrubmuÈ ller24 in a recent 1.3 nscontrol simulation of the monoclonal antibodyAN02 Fab fragment in a stochastic boundary dro-plet containing 13,461 water molecules and 75ions.

Figure 2. Control simulation.(a) Heavy-atom RMSD of the VL

and VH domains (excluding the lin-ker). (b) Radius of gyration of scFv.(c) Solvent-accessible surface asde®ned by Richards.54 (d) Reactioncoordinate: distance between C-2 ofthe ¯uorescein and the C terminusof the antibody. The zero of thetimescale corresponds to the begin-ning of the unconstrained controlrun (i.e. negative times correspondto the 200 ps heating phase and200 ps equilibration with con-strained backbone atoms; see thetext).


The relatively large deviation for the light chainis due to the CDR-L1 loop. This is in accord withthe fact that the most signi®cant discrepanciesbetween the 1.85 AÊ structure29 and the 2.7 AÊ

structure30 are localized in this loop. However, onecannot exclude the possibility that some of thedeviation might originate from the approximationsin the model.

Properties of the antibody during the unbiasedcontrol simulation shown in Figure 2, togetherwith the behavior of the different energy com-ponents (not shown), indicate that the system isequilibrated and stable. The value of the radius ofgyration averaged over the last 800 ps is 18.91 AÊ ,which is close to the value of the initial energy-minimized conformation (18.6 AÊ ). The ligandremains bound to the antibody during the controlrun. The average value of rFA during the last800 ps of the control simulation is 45.3 AÊ with astandard deviation of 1.4 AÊ ; the initial value is43.3 AÊ . The small difference is due primarily to the¯uctuations in the C-terminal segment of the anti-body. The solvent-accessible surface decreasesslightly from an initial value of 11,840 AÊ 2 to anaverage value of 11,480 AÊ 2 during the ®nal 800 ps.Overall, there is a tendency to approach closer tothe crystal structure values in the ®nal stages ofthe equilibration period, suggesting that the modelwould be stable on longer timescales.

The unbinding simulations

Unbinding is induced by adding an external per-turbation that favors the increase in the distancebetween antibody and ligand. Two kinds of time-dependent external perturbations have been used.The timescale for the unbinding is modulated byan adjustable parameter a in the BMD approach(see Materials and Methods) or by the constantexternal force F in an alternative approach where a

Table 1. Overview of the simulations

Starting

Simulationmethod Force constant t � 0.4 t � 0.8

BMD a�0.0001 30 ns 14.8 nsBMD a�0.0002 3.2 ns 7.4 nsBMD a�0.0003 0.64 ns 2.6 nsBMD a�0.0005 0.53 ns 0.79 nsBMD a�0.001 228 ps 179 psBMD a�0.002 126 ps 164 psBMD a�0.003 54 ps 69 psBMD a�0.004 57 ps 60 ps

CFMD F�150 pN 20 nsCFMD F�200 pN 5.2 ns 5.1 nsCFMD F�250 pN 0.21 ns 2.6 nsCFMD F�300 pN 63 ps 226 psCFMD F�375 pN 269 ps 69 psCFMD F�500 pN 34 ps 16 psCFMD F�625 pN 8.4 ps 4.5 ps

In boldface are those that did not result in full unbinding (rFA >molecular dynamics. Initial conformations where taken after t ns of t

constant force is applied. In both cases, severalvalues of the parameters a or F were used to inves-tigate the dependence of the unbinding pathwaysand of the measured properties along the samepathways. For each value of a or F, six simulationswere performed from different bound con®gur-ations (coordinates and velocities extracted at theend of every 0.4 ns interval of the control run).Table 1 lists all the simulations performed: thesewere stopped when unbinding occurred, i.e. thevalue of rFA was greater than 80 AÊ .

In general, a smaller value of a (see equation (8))or of F (see equation (10)) is expected to yield morerealistic results, since the perturbation has a smal-ler effect on the system and the unbinding processis closer to the unbiased one. However, this has tobe balanced by the need to simulate the transitionin a ®nite time. Although the exact kineticsdepends on the initial conditions (i.e. each trajec-tory is somewhat different), it was found that, inthe BMD case, a value a � 0.0002 kcal molÿ1 AÊ ÿ4

(1 cal � 4.184 J; 1 kcal molÿ1 AÊ ÿ1 � 69.4786 pN)induces complete unbinding in less than 8 ns in allthe six simulations. Higher values of the parametera lead to unbinding on a shorter timescale, and thecomparison provides information on the effect of aon the computed quantities. Also, six runs with alower a value (0.0001 kcal molÿ1 AÊ ÿ4) were per-formed for 10 ns or longer and only two led tocomplete separation.

Biased molecular dynamics

With a coupling a � 0.0002 kcal molÿ1 AÊ ÿ4, fullunbinding always occurs in less than 8 ns. Theresults from simulations with larger or lowervalues of a are mentioned as a basis of comparisonand/or to underline important differences. Thereaction coordinate rFA as a function of simulationtime is shown in Figure 3. Once rFA has increased

points on the equilibration trajectory

t � 1.2 t � 1.6 t � 2.0 t � 2.4

10 ns 9.5 ns 10 ns 10 ns3.1 ns 2.5 ns 3.7 ns 2.3 ns2.4 ns 1.1 ns 0.80 ns 2.8 ns3.5 ns 1 ns 0.89 ns 0.57 ns318 ps 299 ps 289 ps 198 ps200 ps 119 ps 75 ps 133 ps80 ps 71 ps 80 ps 68 ps62 ps 55 ps 35 ps 66 ps

2.4 ns 6 ns 1.6 ns 3.4 ns0.97 ns 0.97 ns 4 ns 2.2 ns545 ps 788 ps 384 ps 85 ps156 ps 1 ns 128 ps 88 ps41 ps 55 ps 28 ps 26 ps11 ps 12 ps 7.6 ps 7.7 ps

80 AÊ ). BMD; biased molecular dynamics; CFMD, constant forcehe control run (see the text).


to about 70 AÊ from its initial value near 45 AÊ , theunbinding proceeds without hindrance, since allinteraction terms have gone to zero. In one run,started from the t � 0.8 ns conformation of the con-trol simulation, the rFA distance has increased byonly 10 AÊ after 7 ns, and the unbinding is com-pleted only after continuing the simulation to7.4 ns. It is clear from the trajectories shown inFigure 3, as well as from those for different valuesof a, that unbinding proceeds by a sequence ofrelative discrete jumps. This suggests that there area series of barriers for rFA values between 47 and61 AÊ . The intra-receptor and ligand-receptor inter-actions involved are analyzed below.

The timescale for unbinding has a random com-ponent, as expected for a thermally activated pro-cess. It is not imposed externally as it would be ifthe two ends were pulled at constant speed. This isan important feature of BMD, in contrast to othermethods. In BMD, the effective unbinding speed inthe regions characterized by barriers can be as lowas 0.01 AÊ nsÿ1; however, even this speed is still�103 faster than the fastest experimental pullingspeed (about 10ÿ5 AÊ nsÿ1). A longer search time isallowed when the system encounters a barrier,which makes the motion closer to what might beexpected under the quasi-equilibrium condition ofthe AFM experiment. The values of rFA as a func-tion of time in Figure 3 show several bottleneckregions (i.e. regions where intermolecular inter-actions oppose a further increase of rFA, whichthen remains almost constant for long time inter-vals). For different trajectories, the bottlenecks donot occur at the same time, though they occurmainly at corresponding distances.

A twice smaller value of a (a � 0.0001 kcalmolÿ1 AÊ ÿ4) is not enough to always induceunbinding in 10 ns or longer simulations (seeTable 1). The magnitude of the RMS ¯uctuations in

the reaction coordinate (relative to the runningaverage over 10 ps intervals) is larger witha � 0.0001 kcal molÿ1 AÊ ÿ4 (0.48 AÊ ) than witha � 0.0002 (0.44 AÊ for rFA < 55 AÊ ), and somewhatsmaller than in the control run (0.73 AÊ ). Hence, theperturbation reduces the spontaneous ¯uctuationsof the reaction coordinate by about 35 % to 40 %,when the value of the coupling a is large enoughto observe full unbinding on the nanosecond time-scale. However, the overall atomic ¯uctuations ofthe protein are not altered signi®cantly.

Constant force molecular dynamics (CFMD)

In the BMD approach, the bias is a time-depen-dent perturbation that is non-zero only for displa-cements in the direction that decreases the reactioncoordinate. Although the BMD method has import-ant advantages, as described above, complemen-tary information on the unbinding process can beobtained by switching on a constant force on rFA attime zero to pull apart the atoms F and A (seeMaterials and Methods). This approach has beenused in related studies of the stretch-inducedunfolding of proteins.20,31 Such an external forceadds a term FrFA to the potential energy function.This ``tilts'' the potential energy surface and mod-i®es the shape, magnitude and location of the mini-ma and transition states.32 In fact, in the presenceof a constant force the unbound state becomesenergetically the most favorable, although, depend-ing on the magnitude of the force, the bound statemay remain a local minimum. Unlike BMD, themagnitude of the force is ®xed, so that a forcelarge enough to overcome a barrier situated at avalue of rFA close to the bound state might be toolarge to carefully sample conformations at largerrFA values. However, such a constant force canserve to delineate barriers that cannot be overcomeon the simulation timescale, so that comparison of

Figure 3. The reaction coordinaterFA as a function of time during sixBMD simulations. The couplingparameter a is 0.0002 kcal molÿ1

AÊ ÿ4, which is always suf®cient todissociate the ligand from its sub-strate on a 10 ns timescale. Theinset shows the behavior of rFA inone of the a � 0.0001 kcal molÿ1

AÊ ÿ4 simulations that did not yieldunbinding.


the results obtained with different values of forceare of interest.

The smallest force that induced unbinding was200 pN (®ve simulations over six, in less than6 ns); for this force, Figure 4 shows rFA as a func-tion of the simulation time. In one simulation, thereaction coordinate does not increase aboverFA � 53 AÊ within 6 ns. From the behavior of theother trajectories, an rFA value of about 53 AÊ corre-sponds to one of a series of discrete metastablestates for the potential modi®ed by the constantforce. Bottlenecks around rFA � 58 AÊ andrFA � 66 AÊ are particularly clear in the Figure.With a 150 pN constant force, a 20 ns simulationdid not yield unbinding.

Comparison of results from the twounbinding procedures

To analyze and compare the BMD and CFMDresults, it is useful to consider two different coordi-nates for describing the unbinding process. The®rst is the reaction coordinate, rFA, used in thesimulations and the second is the distance, dCM,between the centers of mass of the ¯uorescein mol-ecule and the antibody, where the seven residuesclosest to the C terminus are disregarded. The lat-ter allows us to focus on the relative motion of the¯uorescein and the ligand, without the protein dis-tortion that contributes signi®cantly to rFA. InFigure 5, histograms are plotted for the times (inps) during which the rFA values were in givenintervals for the BMD and CFMD simulations; theresults correspond to a sum over all the simu-lations of a given kind (i.e. with a given value of aor F). BMD results for the trajectories with abetween 0.0001 and 0.002 kcal molÿ1 AÊ ÿ4 areshown in Figure 5(a); analogous results for the

CFMD trajectories for F between 150 and 500 pNare shown in Figure 5(b). For the number of trajec-tories used in the averages, see Materials andMethods.

The histograms provide information that ampli-®es the analysis of the varying slopes seen in theunfolding diagrams in Figures 3 and 4. ChoosingrFA as the progress variable, the BMD trajectories(Figure 5(a) show a probability distribution thatvaries signi®cantly as a increases. At the smallest avalue (a � 0.0001 kcal molÿ1 AÊ ÿ4) there is a singlewell-de®ned peak at 52 AÊ (bottleneck) that movesinwards as a increases to 0.0003 and then disap-pears. Simultaneously, a second peak appears ata � 0.0002, which increases in probability andmoves inward at a � 0.0005. Finally, at the largesta value (0.002), there is a broad distribution of resi-dence times, which suggests that there is no majorbarrier; i.e. the BMD perturbation is so large thatthe intrinsic potential of the system no longer playsa signi®cant role. The constant force result(Figure 5(b)) reveals a similar transition betweendifferent regimes. One occurs at forces below300 pN, where the most populated state is aroundrFA � 50 AÊ , and another one at forces larger than300 pN where the most populated state is aroundrFA � 60 AÊ . At 300 pN, the longest-lived intermedi-ate is present between rFA � 50 and 54 AÊ .

The above results correspond to the differentways in which the effective energy surface is per-turbed in the two approaches: BMD introduces abias that locally ``¯attens'' the energy surface sothat the reaction coordinate increases rathersmoothly as barriers are crossed, while CFMDglobally tilts the energy surface, increasing theprobability of jumps over the barriers. If the pro-cess of unbinding could be described effectively asthe crossing of multiple barriers along a one-

Figure 4. Time behavior of thereaction coordinate during sixCFMD simulations. In ®ve casesout of six, a 200 pN force is suf®-cient to dissociate the ligand fromits substrate completely in less than6 ns.

Figure 5. (a) Frequency of occurrence of rFA for theBMD simulations with 0.002 4 a 4 0.0001 kcal molÿ1

AÊ ÿ4 and (b) for the constant force trajectories with150 4 F 4 500 pN.


dimensional reaction coordinate rFA, a constantforce would decrease the outer barriers more thaninner barriers (see below), possibly displacing theposition of the main transition state and affectingthe unbinding pathway. This is the suggested ori-gin of the different regimes observed experimen-tally by Merkel et al.33 for the forced unbinding ofstreptavidin from biotin. In our simulations, weobserve a more complex behavior. There are sev-eral regimes, in each of which the barrier movesinward as the perturbation increases, while in thedifferent regimes the barrier moves outward as theforce increases.

Unbinding forces in the BMD trajectories

Figure 6 shows the magnitude of the force as afunction of rFA for the six unbinding trajectorieswith a � 0.0002 kcal molÿ1 AÊ ÿ4; the force was cal-culated by block averaging over intervals of 2 AÊ inrFA, since there are large instantaneous ¯uctuationsthat obscure the overall behavior. The force pro®lesas a function of rFA for the different trajectories aresimilar, though not identical. In all cases, the forcerises quickly as rFA increases by only a few aÊng-stroÈm units relative to the initial value. The forcetends to have an absolute maximum of about250 pN for rFA values between 48 and 54 AÊ . Insome cases, the force peak is broader (48 to 64 AÊ )and certain trajectories have secondary peaks(about 150-200 pN) at rFA values between 63 and72 AÊ . The force decreases to zero at rFA between 75and 80 AÊ , i.e. at the distance where the hapten andthe scFv fragment are separated completely.

Figure 7 shows the unbinding forces averagedover the BMD trajectories (Table 1) at different avalues. The overall shapes of the force curves aresimilar, with a major broad peak between 50 and60 AÊ , where the magnitude increases for increasinga, as expected. The results are compared with themean force below.

Mean force along the reaction coordinate

Since BMD is expected to yield pathways thatare close to equilibrium when a low value of a isused, it is of interest to determine the mean forcealong the reaction coordinate (equation (11)). Themean force pro®le was obtained by using the struc-tures calculated during the simulation as startingpoints for runs with rFA constrained to differentvalues. Speci®cally, the initial phase points (pos-itions and velocities) for the constrained trajectorieswere taken either from the control run (forrFA � 45.8 AÊ ) or from the BMD unbinding runs.Simulations with rFA constrained to 49.9, 55.2, 59.4,70.2, 75.6 and 78.8 AÊ , each for 1 ns, were per-formed to calculate the ®rst term on the right ofequation (11); in general, it converges within the®rst 200 ps. The second term in equation (11),which is always less than �2 pN, was neglected.The calculated mean force is shown in Figure 7and is compared to the average external forcesobtained during BMD unbinding simulations withdifferent a values. Both the magnitudes of theforces and their dependence on rFA are similar.Moreover, for rFA < 65 AÊ the BMD force for thelowest a that leads to unbinding (a � 0.0002 kcalmolÿ1 AÊ ÿ4) gives results closest to the mean force,while slightly larger forces are obtained for largera. This supports the conclusion that the initial con-formations for the mean force calculations,although extracted from BMD trajectories, are notfar from the equilibrium unfolding pathways. Thestandard deviation for the mean force computedby averaging over 2 ps blocks ranges from 1.1 to5.4 pN. Use of larger blocks suggests that the error

Figure 6. Force due to the exter-nal perturbation as a function ofthe distance for the six trajectorieswith a � 0.0002 kcal molÿ1 AÊ ÿ4.


might be underestimated in this way, as is oftenthe case in this type of analysis; e.g. for the trajec-tory with rFA constrained at 55.2 AÊ , the differencebetween the averages over the ®rst and the secondhalf nanosecond is about 19 pN, compared to themean force of 352 pN. Although this indicates alarger error, it is still small relative to the totalmean force.

The mean force shows smooth behavior as afunction of rFA, except for the point at rFA � 59.4 AÊ .This is lower than might be expected from theother values, though not in disagreement with thedouble peak observed in most BMD results with

Figure 7. Mean force (diamonds) obtained from eight Mmations from BMD unbinding runs (see the text). Simulationfor 2 ns. At rFA � 59.4 AÊ , the three values of the mean force(diamond), the ®rst 700 ps (open circle), and the last 700 psbecause the mean force jumps from a high to a low value dresent the external force averaged over the BMD trajectories

a 5 0.001. Analysis showed that for rFA con-strained to 59.4 AÊ , the force has an average valueof 409 pN during the ®rst 0.7 ns and 47.3 pNduring the last 0.7 ns (from 1.3 to 2 ns). An exam-ination of the constrained trajectory revealed thatduring the ®rst 0.7 ns the ¯uorescein ligand is bur-ied in the antigen-binding site, while the C-term-inal portion of the scFv fragment (from Gly255 toSer262) is completely stretched and separated fromthe rest of the antibody (see also below). During0.7 ns to 1.3 ns, the ¯uorescein ligand leaves thebinding site, and after 1.3 ns it is almost completelyunbound except for the oxygen atoms of the three-

D simulations using equation (11), started from confor-s were 1 ns long except for rFA � 59.4 AÊ , which was runcorrespond to the average over the whole 2 ns trajectory(open triangle). The latter two averages were computeduring the 2 ns simulation (see the text). The curves rep-

at the four smallest values of a used (see Table 1).

Figure 8. Components of the non-bonded potentialenergy of the whole system. The diamond representsthe corresponding value over the last 800 ps on the con-trol run, while the error bar is the RMS ¯uctuation ofthe corresponding quantity within the 3 AÊ interval ofrFA over which it is averaged. (To make comparisonsimple, the scale is the same in all the windows of theplot).


ring system, which still accept hydrogen bondsfrom the side-chains of His31, Gln33, and the mainchain NH of Asn35. The departure of ¯uoresceinfrom the binding site is concomitant with re-associ-ation of the C-terminal residues with the rest of thescFv fragment. A parallel b-sheet involving resi-dues 150-154 and 257-261 is formed; it is shifted bytwo residues with respect to the native parallel b-arrangement. This indicates that the effective bar-rier to breaking the contacts between ¯uoresceinand the binding site residues is high, in accordwith the large force required. For rFA valuesbetween 45 and 55 AÊ , it is apparently easier toincrease the value of the reaction coordinate by thedisplacement and partial stretching of the C-term-inal residues than by unbinding of the ¯uorescein.The time-dependence of distance dCM between thecenters of mass of the antibody and the ¯uoresceinclearly re¯ects this result. For the ®rst 0.7 ns, dCM isabout 16 AÊ , and between 0.7 and 1.3 ns dCM

increases to about 27 AÊ in three steps. The con-strained simulation at rFA � 59.4 AÊ indicates thatthe initial conformation is not an equilibrium one,even on the nanosecond timescale. This empha-sizes the fact that the unbinding takes place in amany-dimensional space and that use of a singlereaction coordinate involves a projection. For mostof the unbinding trajectories, rFA is a meaningfulchoice. However, at rFA � 59.4 AÊ , a second dimen-sion is required. While the reaction coordinate rFA

is kept constant, dCM changes signi®cantly andresults in a large variation of the mean force overdifferent segments of the simulation with rFA ®xed.The choice of dCM as reaction coordinate wouldpossibly improve the convergence because the sys-tem has less freedom to relax when dCM, and notrFA, is constrained to a ®xed value. However, wehave chosen rFA as the reaction coordinate, becauseit corresponds to that used in the experiment forthe anchoring sites of the antibody and ¯uoresceinto the AFM device. It is clear that the slower theforced unbinding is, the more critical is the role ofthe relaxation in a direction orthogonal to the cho-sen reaction coordinate; i.e. in the present case,that relaxation is on the nanosecond timescale ofthe simulation. Comparing Figures 6 and 7, it isevident that corresponding events in the differentsimulations happen for slightly different values ofrFA even at a given a value. Thus, in the averageover a set of trajectories, the barriers at differentpositions appear as a single broad barrier(Figure 7). Although the AFM experiments are ona much slower timescale (millisecond to second),corresponding relaxation has been observed insingle-molecule experiments,34 as well as in time-resolved photodissociation experiments.35 It is clearthat protein relaxation covers many orders of mag-nitude in time that extend into the second range.Thus, processes along directions orthogonal to thepulling coordinate are likely to complicate theexperimental result.

Except for the trajectory with rFA constrained at59.4 AÊ , convergence is achieved on the nanosecond

timescale, as monitored by relevant parameters.The total effective energy is essentially constantduring the constrained simulations, and the RMSDof the protein from the starting con®gurations foreach rFA converges quickly to low values, between2 and 4 AÊ for heavy atoms (without the linker).This is in accord with the conclusion that confor-mations extracted from the biased unbinding are,in most cases, similar to equilibrium conformationsobtained with a constraint on the reaction coordi-nate. In other words, the non-equilibrium samplingprovides an estimation of the conditional prob-ability of a conformation, given a value of rFA.

Energetics of unbinding

Figure 8 shows the total effective energy and thenon-bonded energy terms (van der Waals, electro-static, and solvation, see Materials and Methods)as a function of rFA for the six runs witha � 0.0005 kcal molÿ1 AÊ ÿ4; the values correspondto averages over intervals of 3 AÊ . Although the¯uctuations are large, the total energy can be seento increase by approximately 100 kcal molÿ1. Thisis due to increases in both the van der Waals andelectrostatic interactions, which are not compen-sated completely by the decrease in the solvationenergy. The incomplete compensation is due pri-marily to the exposure to solvent of the apolar sur-


face of ¯uorescein during unbinding. The bondedterms are almost constant along the unbindingpathway (not shown). We note that this analysis iseasy to do only with the implicit solvation model,which provides a value, albeit approximate, for thefree energy of solvation. By contrast, the estimationof solvation free energies from explicit water simu-lations would require very costly evaluations ofthe potential of mean force along the unbindingpathways.

The protein energy, which has large ¯uctuations,is correlated, for rFA values up to about 70 AÊ , withthe RMSD from the initial structure. This suggeststhat a moderate increase (30 kcal molÿ1) in the pro-tein energy arises from the deformation, whichprecedes the breaking of the ligand-receptor con-tacts (see above). The ligand energy is not alteredduring and after unbinding because of its rigidity.

Figure 9 shows the interaction energy betweenthe ligand and the receptor as a function of rFA.The value of rFA (but not the distance between¯uorescein and the hapten-binding site) increasesby 10 AÊ up to about 55 AÊ without any noticeabledifference in the intermolecular energy. This is dueto the displacement of the extended strand consist-ing of the seven C-terminal residues of the scFvfragment that was described above, without anysigni®cant change in the conformation of the bind-ing site and the binding mode of ¯uorescein; thedistance dCM is essentially at its equilibrium value.The intermolecular energy is about ÿ70 kcal molÿ1

in the native bound conformation and goes to zeroin the unbound state at �80 AÊ , the distance atwhich ligand and antibody no longer interact. Thelargest change takes place at rFA values between 55and 57 AÊ because of the sudden loss of two to fourintermolecular hydrogen bonds and favorable van

Figure 9. Fluorescein-antibody interaction energy (vander Waals, electrostatic and their sum) as a function ofthe reaction coordinate. Each point represents an aver-age over all the con®gurations from the unbinding tra-jectories for rFA in a 1 AÊ interval, and error barsrepresent the magnitude of ¯uctuations.

der Waals interactions between ¯uorescein and anumber of aromatic side-chains (see the nextsection).

Structural analysis of the unbinding

We ®rst describe the sequence of events for theruns with small values of the bias parameters a orF. For rFA values up to about 55 AÊ the position andorientation of ¯uorescein in the antigen-bindingsite varies only slightly. The increase of about 10 AÊ

in the rFA distance (from 45 to 55 AÊ ) is primarilydue to a displacement of the extended strand con-sisting of the seven C-terminal residues of the scFvfragment (Figure 10), without any signi®cantchange in the binding mode or interaction energyof ¯uorescein. The C-terminal strand moves awayfrom the rest of the antibody as a consequence ofthe force acting on it; this results from the choice ofthe reaction coordinate, which corresponds to thepoint of attachment of the antibody in the AFMexperiment. Since the scFv fragment consists ofonly the two variable domains, it is possible thatthe ¯exibility of its C-terminal residues is due tothe lack of the stabilizing interactions providedby the constant domains in the complete Fabfragment.

To eliminate the complication of the C-terminalend we consider the distance dCM between the cen-ter of ¯uorescein and the protein as the reactioncoordinate. As dCM increases from d � 14 AÊ to17 AÊ , the ¯uorescein is slowly moving out of thepocket with certain aromatic residues (Tyr37,Trp177, and Tyr247) and hydrogen bondinggroups (Ser96 and Arg196) following the ¯uor-escein. When dCM is between 17 and 19 AÊ , most ofthe trajectories undergo a sudden transition, whichincreases dCM by 2 to 4 AÊ . For d < 19 AÊ , there is atendency to unbind rapidly. An analysis of the tra-jectories reveals that for dCM between 17 and 19 AÊ

the hydrogen bond between the backbone CO ofSer96 and the OH of ¯uorescein breaks ®rst, fol-lowed by that between the guanidinium group ofArg196 and the ¯uorescein carbonyl group. Thebreaking of the hydrogen bonds at the transitionstate is in accord with the interaction energy results(Figure 9), which shows a sudden decrease in mag-nitude for both the van der Waals and electrostaticcontribution at rFA � 55 AÊ .

What we have described is a ``consensus'' path,which corresponds to the main features of mostunbinding trajectories in the simulations withweak bias parameters, although there are somedifferences among them. This variation is due tothe fact, already mentioned, that the details of theunbinding cannot be described in terms of a singlecoordinate (either rFA or dCM), but that a more com-plex progress variable, including other coordinates,is required for a complete description. Such com-plexity exists even in the ``simple'' case of dihedralangle transitions in the alanine dipeptide, where abroad transition region was found when the tran-sition is expressed in terms of the variables f and

Figure 10. Stereoplot (wall-eye) of the unbinding simulation started at t � 0.4 ns with a � 0.0005 kcal molÿ1 AÊ ÿ4.Snapshots are shown for conformations saved at 1 ps (rFA � 46.5 AÊ ), 433 ps (55 AÊ ), and 518 ps (70 AÊ ). The backboneatoms of the scFv fragment are colored in green (VL), blue (VH), and black (linker). Fluorescein is shown in red. TherFA distance is shown as a red line. The Figure shows that the most ¯exible parts of the scFv fragment are the linkerand the C-terminal segment (see the text).


c.36 Nevertheless, the important interactions areapparent from the present analysis.

The simulations with a > 0.0005 or F > 300 pNshow a different sequence of events. The value ofrFA increases to 55 AÊ (and that of dCM to about19 AÊ ) because of the almost complete exit of ¯uor-escein from the binding site without any detach-ment of the C-terminal strand. In most of theseruns, the ¯uorescein molecule is kept at theentrance of the binding site (i.e. dCM of about 19 AÊ )by two hydrogen bonds: between its carbonyl oxy-gen atom and the guanidinium group of Arg196,and between its hydroxyl group and the main-chain carbonyl group of Ser96. During the laststeps of unbinding (rFA > 55), the seven residues ofthe C-terminal strand separate from the rest of thescFv either partially (i.e. rupture of only the twohydrogen bonds involving residue 156 on onestrand, and the CO of residue 259 and the NH ofresidue 261 on the other strand) or totally. Hence,the sequence of events under a strong bias is differ-ent than at low bias. The rate-limiting step seemsto involve, in both cases, the separation of the C-terminal strand from the rest of the scFv. This hap-pens before the exit of ¯uorescein from the bindingsite (dCM < 17) at low values of the bias parameter,and after ¯uorescein has undergone a signi®cantdisplacement from its bound position (dCM > 19) athigh bias. This partially explains the two differentregimes in Figure 5.

Estimation of unbinding rates

A bound state can be idealized as con®ned by asingle energy barrier at a given position along aspeci®c reaction path. Assuming that the kineticsof the forced unbinding can be analyzed in thecontext of the Kramers theory for activatedprocesses,32,37 the off-rate for unbinding can bewritten as:

koff � oeÿb�Ey �1�where b � 1/kBT, kB is the Boltzmann constant, Tis absolute temperature, and o is a frequencyprefactor. The activation energy �E{ is equal toE(rFA

{ ) ÿ E(rFAbound) and { indicates the transition

state.In the AFM experiments, an external force is

applied to a pair of atoms that are pulled apart.Since the timescale of thermal motion and the time-scale of the force variation in the AFM experimentare separated by at least eight orders of magnitude,the assumption is usually made that the effect ofthe external force on the microscopic kinetics isthat of a constant force. As ®rst argued by Bell,38 aconstant force F applied to rFA modi®es the energy(and the free energy) by a factor FrFA. Thus, if Fand rFA are parallel:

�Ey�F� � �Ey ÿ FryFA �2�which gives a rate for unbinding that depends on

Figure 11. Unbinding force as a function of theunbinding rate in the BMD and CFMD simulations.


the applied force:

koff�F� � koffebFry

FA �3�if one assumes (as is commonly done) that koff isgiven by equation (1) and that the prefactor o isnot affected by the force.

Experimentally, the average unbinding force (theheight of the peak) as a function of pulling speedis usually measured. With a series of (reasonable)assumptions it can be shown32,39 that the mostprobable unbinding force is given by:

F � 1

bryFA

lnbrryFA

koff

!�4�

where r is the loading rate, de®ned as the timederivative of the force applied to the bond. Theloading rate r is assumed to be equal to the pro-duct of the force constant of the AFM cantileverand its retraction speed. It is equal to the true load-ing rate r only if the ``bond'' is ®xed in position.This assumption was used to analyze the exper-imental AFM results13 for the same ligand-anti-body system as studied here.

In the simulations two different methods (BMD,using various a values and CFMD using various Fvalues) were used to study the unbinding. Theresults obtained for multiple trajectories are sum-marized in Table 1. As can be seen, for each a or Fset that results in complete unbinding, a range ofvalues for the unbinding times is obtained. Onedifference between BMD and CFMD is that theunfolding time varies in a narrower range at agiven value of a than at a given value of F. Thisdifference between the results of the two methodssupports the choice of BMD, which is more compli-cated to interpret but somewhat more ef®cient.Approximate average unbinding times and rates(the inverse time) can be computed and extrapo-lated to a � 0 or F � 0. These limits correspond tono applied force, when the unbinding rate shouldapproach the off-rate in solution, koff, if the simplemodel just described is applicable. From the resultsgiven in the previous section, including the multi-dimensional character of the ``true'' reaction coor-dinate, this is likely not to be true. Nevertheless, itis of interest to determine the dependence of therequired unbinding force on the unbinding rate.

Equation (3) can be used to estimate koff andrFA{ from the CFMD and BMD simulations. In the

latter case, the unbinding force is estimated as themaximum of the bias force F(rFA) block averagedover intervals of 1.5 AÊ in rFA. Rewriting equation(3) as:

F � 1

bryFA

lnkoff�F�

koff

� ��5�

the off-rate at zero force can be estimated as theintercept at F � 0.

Figure 11 shows the unbinding force as a func-tion of the unbinding rate in the BMD and CFMDsimulations. The plots of F as a function of ln(k)show a linear behavior (correlation r2 ' 0.99).The results in Table 1 give koff � 5.8(�0.7) � 10ÿ6

psÿ1 and rFA{ � 0.73(�0.05) AÊ (BMD) and

koff � 2.4(0.4) � 10ÿ5 psÿ1 and rFA{ � 0.58(�0.05) AÊ

(CFMD). The two sets of results are the samewithin the uncertainty of the calculations; a sourceof systematic error could be that koff is estimated atlow forces by disregarding the trajectories that donot lead to unbinding.

The above results for the two parameters of thesimple model are very different from the exper-imental estimates, koff(0) � 0.1 sÿ1 and rFA

{ � 4 AÊ .13

This could be due to the fact that simulated forcedunbinding proceeds through a pathway that isdifferent from that explored in AFM unbinding.More likely, the effective unbinding rates of theforced simulations might be determined by innerbarriers (i.e. at lower rFA

{ ) that become dominantunder a large externally applied force. It is interest-ing to note that Heymann & GrubmuÈ ller24 foundfor a similar system that the unbinding length (i.e.the displacement of the two anchoring site at therupture point) varies between 0 and 8 AÊ , while thevalue obtained independently from the ®t was2.2 AÊ . The present results also indicate that, whilethe two-state model is appropriate to ®t thenumerical results (i.e. the dependence of theinverse average unfolding time of the externalforce is logarithmic), the rFA

{ parameter appears notto be the true ``position'' of the TS in terms of therFA coordinate. The same simulations from whichwe derived such a value for the ®tting parameterrFA{ showed, in fact, that the ¯uorescein departure

from the binding site occurs in steps over a lengthof several aÊngstroÈm units. This provides a caution-ary note concerning the interpretation of the exper-imental data; i.e. the rFA parameter obtained from


the experiment may not have a real physical mean-ing.

Using the value of �E{ � 95.9 kcal molÿ1

reported by Schwesinger et al.13 and the relation:

kexpoff

ksimoff

� 10ÿ1 sÿ1

107 sÿ1� eÿb��Eyexpÿ�Ey

sim� �6�

we derive �E{sim � �E{

exp � 0.6ln 10ÿ8 � 95.9 ÿ 11kcal molÿ1 � 84.9 kcal molÿ1; assuming that onlytwo barriers are present, located at rFA � 0.7 and4 AÊ , respectively, the crossover between inner andouter barriers happens at a force equal to 11/(4 ÿ 0.7) kcal (AÊ mol)ÿ1 � 230 pN. However, theresults reported above suggest that the interpret-ation of values obtained by assuming that the sys-tem is a two-state system following a uniquepathway that can be projected onto the variablerFA might be an oversimpli®cation.

Concluding Discussion

The forced unbinding of ¯uorescein from the4D5Flu scFv fragment was investigated using twotypes of external forces and exploring a broadrange of timescales by molecular dynamics with animplicit solvent. In the simulations, the force-®eldwas supplemented by either a time-dependentbiasing perturbation or a constant pulling force.The former scheme allows one to exploit the spon-taneous ¯uctuations of the system. We ®nd evi-dence for the importance of comparing differentmethods for inducing the unbinding on a timescaleaccessible to the simulation, and for exploring abroad range of timescales.

The present simulation results indicate that theunbinding pathways are close to equilibrium inspite of the very high effective pulling speed.Although the unbinding simulations are performedat a pulling speed at which a linear dependence ofthe unbinding force on the pulling would beexpected due to Stokes' friction,23 the absence ofexplicit water is expected to reduce this effect; onlythe internal friction, which is expected to be small,and the force contribution related to the activatedcrossing of the free energy barriers to unbindingremain. The derivative of the potential of meanforce (i.e. the mean force) computed over theunbinding pathways shows good convergence ona sub-nanosecond timescale. Also, the mean forcepro®le is similar to the externally applied forceaveraged over the unbinding simulations.

Experimentally, it has been found recently13 thatthe unbinding force depends linearly on the logar-ithm of the loading rate and that by an extrapol-ation to zero force the off-rate in solution can bedetermined. This was interpreted as a demon-stration of the fact that, in the range of loadingrates explored (the loading rate is the product ofthe force constant of the AFM cantilever and itsretraction speed), the unbinding forces of the scFvfragment-¯uorescein system are independent of

the loading rate and thus determined by the sameenergy barrier as in solution. In the range of load-ing rates going from zero to 104 pN sÿ1 the sametransition state is found for the unbinding. This issimilar to what found by Merkel et al.33 for strepta-vidin unbinding from biotin. However, in thatcase, at larger loading rates, linear relations withdifferent slopes resulted between force and the log-arithm of the loading rate, suggesting that differenttransition state(s) are explored at larger loadingrates.

In the present simulations, number of quantitiescomputed along the unbinding trajectories suggestmultiple pathways that depend on the externalperturbation and the timescale over which theunbinding is induced. However, of the structuraldetails and the breaking of individual atom-atominteractions during the unbinding show that thesepathways follow a well-de®ned sequence of events.These are shown to involve the whole antibody,not only the binding site. In particular, we observea distortion of the scFv fragment that is localizedmainly at the C-terminal region (residues 255-262)and are partially reversible, while ¯uorescein is notdeformed during unbinding. This ``compliance'' ofthe antibody, involving a reversible deformation, isoften ignored in the analysis of AFM data.

The structural details of the unbinding haverevealed that most of the residues involved in theinteractions with ¯uorescein, which are responsiblefor the peak of the force, are the same in all simu-lations. The peak of the force precedes the loss ofthree to ®ve intermolecular hydrogen bonds andthe rupture of favorable van der Waals contactswith the aromatic side-chains of the hapten-bind-ing site (Tyr37, Trp177 and Tyr247). It would be ofinterest to mutate these residues to determinewhether any of them modulate the unbindingforce.

Materials and Methods

Simulation system

The 4D5Flu scFv fragment consists of the complemen-tary determining regions (CDRs) of the ¯uorescein-bind-ing antibody 4-4-20 transplanted into the framework ofthe humanized 4D5Flu antibody.14 The loop graftingimproved the stability and folding yield relative to 4-4-20, an antibody with very high aggregation tendency,40,41

which yields almost no soluble expression in the peri-plasm of Escherichia coli. The 4D5Flu scFv fragment con-sists of two variable domains VL (residues 1-114, or L1-L109 in Kabat42 numbering) and VH (residues 145-262, orH1-H113 in Kabat numbering) connected by the ¯exiblelinker (Gly-Gly-Gly-Gly-Ser)6. The scFv 4D5Flu has beencharacterized by a number of biophysical techniques.Fluorescence quenching data have shown that it has abinding af®nity for ¯uorescein (Kd � 2.2 � 10ÿ8 M) ingood agreement with that of scFv 4-4-20(Kd � 2.33 � 10ÿ8 M);14 the on and off-rate in solutionhave been measured.13 A model of the 4D5Flu scFv-¯u-orescein complex was constructed by Jung & PluÈ ckthun14

by superposing its VL and the VH domains with the crys-tal structures of the Fv 4D5 (PDB entry 1fvc) and Fab


fragment 4-4-20 (PDB entry 4fab) and checking the CDR-framework contacts. The simulations reported here makeuse of an improved model (A. Honegger and A. P.,unpublished results) based on the 1.85 AÊ resolutionstructure of the 4-4-20 Fab bound to ¯uorescein29 (PDBentry 1¯r). A ribbon representation of the model of thecomplex is shown in Figure 1.

Force-field and implicit solvent model

Molecular dynamics simulations were performed witha version of the CHARMM program43 modi®ed toinclude the biasing perturbation that accelerates theunbinding process (see below). A polar-hydrogen modelwas used for the protein18 and a continuum Gaussianmodel for the solvent.19 A cut-off on long-range inter-actions (group-based switch between 7 and 9 AÊ ) consist-ent with the solvent model parameters was used. Theimplicit solvent model leads to a simulation time similarto that required for an in vacuo calculation, so thatenough simulations can be carried out rather easily toobtain statistically meaningful results. Moreover, it pro-vides a potential-of-mean-force description of thesolvent44 that is appropriate in view of the experimentalpulling speed, which is many orders of magnitudeslower than the solvent relaxation time.

CHARMM43 atom types were used to describe the ¯u-orescein molecule; they are indicated in Figure 1(c),which also gives the partial charges of the ¯uoresceinatoms. The latter were derived as follows. A formalcharge of ÿ1 was assigned to the carboxyl group, whichhas a pKa of 2.2; the pH of the experiment is 7.4.7 A pre-liminary set of partial atomic charges was determined bya method based on modi®ed partial equalization of orbi-tal electronegativity (MPEOE).45,46 The MPEOE approachyields partial charges that depend only on the molecularconnectivity and not on the conformation. These weremodi®ed by using the implementation of MPEOE in theprogram WITNOTP (A. Widmer, Novartis Pharma,Basel, Switzerland, unpublished), which introduces con-straints on the partial charges based on proteinaceousfragments in the MSI CHARMm all-hydrogen force-®eld(Molecular Simulations Inc., Burlington, MA, USA). Theconformation of ¯uorescein was obtained from the Cam-bridge Crystal Data Bank47 and minimized with the MSICHARMm all-hydrogen force-®eld and MPEOE partialcharges. The heavy-atom root-mean-square deviation(RMSD) between the minimized structure of ¯uoresceinand the conformation in the complex with the antibody(PDB ®le 1¯r) is 0.24 AÊ . The only signi®cant difference isthe orientation of the phenyl ring which is almost per-pendicular to the plane of the three-ring system in theminimized conformation (angle of 80 �), while in the ¯u-orescein complexed to the antibody the angle is 65 �. Theminimized conformation was used for a single-point abinitio calculation at the HF/31-G* level of theory withGaussian94,48 and the ®nal partial charges were obtainedfrom a Mulliken population analysis of the HF/31-G*wave function. In the polar-hydrogen model the arylhydrogen atoms are not considered explicitly; hence, foreach CH group the partial charge of the hydrogen atomwas added to that of the bonded carbon atom. The for-mal charge on the carboxyl group was neutralized byincreasing the polarity (qC � 1.35 and qOC � ÿ 0.60, seeFigure 1(c)) to be consistent with the solvation model forthe ionic side-chains of the protein.19 All parameters forbonds, angles and dihedrals of ¯uorescein are equal tothose of the corresponding CHARMM atom types.

Setup and control run

The model structure was ®rst minimized with 1000steps of the steepest-descent algorithm. The local mini-mum was very close to the initial structure; the RMSD is0.31 AÊ for all heavy atoms and 0.22 AÊ for Ca atoms. Thesystem was then heated from 0 to 300 K in 15 K steps byscaling up the velocities by a single factor every 4 ps(80 ps in total). The system was then equilibrated for120 ps by rescaling the velocities when the temperaturedeviated from 300 K by more than 30 K. During the200 ps, all the backbone atoms were constrained toremain near to their initial positions by a harmonic force.The simulation was continued in the canonical ensembleusing the NoseÂ-Hoover thermostat49,50 for 200 ps; all thebackbone atoms except those of the linker (residues 115-144) were restrained harmonically. This was followed bya 2.4 ns unconstrained simulation at 300 K (control run)without rescaling of the velocities. Initial con®gurationsfor unbinding simulations were selected each 400 psfrom the control run to insure statistical independence.A time-step of 2 fs was used and coordinates were savedevery 500 steps (1 ps).

Biased molecular dynamics (BMD)

The biasing force used to unbind ¯uorescein from theantigen-binding site of the 4D5Flu scFv fragment is atime-dependent perturbation proposed by Ballone &Rubini (unpublished) for studying the crystallization kin-etics in model systems. A particular implementation ofthe technique, inspired by an analogy with the AFMexperiments, has been used to simulate the stretch-induced unfolding of a number of proteins.20,21 In thepresent case, in accord with the AFM experiments byPluÈ ckthun and collaborators,7,13 the reaction coordinateleading from the initial to the ®nal state was chosen tobe:

r�t� � r2FA �7�

where rFA is the distance between C-2 of ¯uorescein (F)and the carboxyl carbon atom of the C-terminal residue(SerH113) of the scFv antibody fragment (A); seeFigure 1. Use of the latter as one end of the reactioncoordinate was dictated by the experimental method,7,13

in which the scFv protein was immobilized on a goldsurface through a spacer and a single free thiol group,which extends the C terminus. This choice may beimportant, since it in¯uences the strain in the interior ofthe scFv fragment during unbinding. By contrast, Hey-mann & GrubmuÈ ller,24 who pulled a hapten out of anantibody by a somewhat different molecular dynamicssimulation procedure, used a coordinate correspondingto the distance between the center of mass of the proteinand the ¯uorescein.

To force the system to sample regions of the con®gur-ation space that are separated by either thermodynamicor kinetic (on the simulation timescale) barriers, the mol-ecular effective energy function was supplemented by aperturbation of the form:

W�r; t� �� a

2�rÿ ra�2 if r�t� < ra

0 if r�t�5ra

�8�

where ra(t) is the maximum value reached by r duringtime 0 to t and a is the force constant for the biasing per-turbation. The choice of rFA

2 rather than rFA as the reac-


tion coordinate means that the force is less for small rFA,the region of most interest.

The perturbation is asymmetric in the sense that isquadratic or zero according to the sign of r(t) ÿ ra anddepends on the time through the reaction coordinate.The simulation is started at t � 0 and the value of ra(0) isset equal to r(0), the square of the C2/Flu-C/Ser H113distance (rFA � 42.3 AÊ ) of the equilibrated starting con-®guration. If this distance increases spontaneously in thesimulation step from t to t � �t, i.e. r(t � �t) > ra(t), theexternal perturbation is zero and has no effect on thedynamics. In such a case, ra(t) is updated and W(r, t) ismodi®ed accordingly; i.e. ra(t) is set equal to r(t � �t). Ifr(t) is smaller than ra, the harmonic force (see equation(8)) acts on r to prevent the reaction coordinate fromdecreasing signi®cantly; the value of a determines themagnitude of the backward ¯uctuation of the reactioncoordinate that can occur. We refer to a moleculardynamics simulation with this type of time-dependentperturbation added to the Hamiltonian as biased mol-ecular dynamics (BMD).

In BMD, the macroscopic state of the system is neverchanged, since the perturbation is numerically zerowhen it is added to the Hamiltonian of the unperturbedsystem. Nevertheless, the perturbation affects the evol-ution of system; i.e. it acts like a Maxwell demon in thatit drives the system in a certain direction by ``selecting''the sign of the spontaneous ¯uctuations of a given func-tion of the internal coordinates. If the effective energysurface is such that the motion of the reaction coordinateis diffusive in the absence of a barrier, the temperatureof the system is not expected to change during the con-formational transition. However, if there is a barrieralong the reaction path, the effect of the directed motioninduced by the perturbation is to transform some of thekinetic energy associated with the reaction coordinateinto potential energy. This would lower the temperaturein an isolated system. To avoid possible artifacts fromtemperature variations of this type, the simulations wereperformed at constant temperature using a NoseÂ-Hooverthermostat. In studies of protein unfolding,21 it wasdetermined that no signi®cant local heating is introducedby this method.

In the BMD method, the force on the system varieswith time. The external perturbation corresponds to apair of forces (identical in magnitude and pointing inopposite directions) applied to the C terminus of thescFv fragment and C2 of ¯uorescein and directed alongthe vector rFA joining the A and F atoms; that is:

FWF � ÿFW

A ��ÿ2a�rÿ ra�rFA if r�t� <; ra

0 if r�t�5ra�9�

Whenever a spontaneous ¯uctuation of the system tendsto decrease rFA, the external force acts to pull the F andA atoms apart.

The average force for given intervals of the reactioncoordinate is larger if the system is crossing barriers andsmaller (or equal to zero) if the system is moving freelyin the direction of the reaction coordinate. For compari-son, a series of simulations were performed with a forceof constant magnitude.

In this case, the potential energy term:

W � FrFA �10�was added to the Hamiltonian. This introduces a pair ofconstant forces acting on the F and A atoms; the forces

are identical in magnitude and point in opposite direc-tions along the rFA vector.

The use of the implicit representation of the solvent51

made it possible to perform many simulations of 1 ns orlonger (50 hours of CPU time on a Pentium II 400 MHzprocessor are required for a 1 ns run).

Calculation of mean force

To compute the mean (equilibrium) force along thepathway determined by BMD, molecular dynamicssimulations with the reaction coordinate constrained to agiven value were performed. The mean force along thereaction coordinate rFA can be written25,52 as:

F�~rFA� � ÿ @V

@rFA

� �~rFA

� 2

b~rFA�11�

where b � 1/kBT. The ®rst term is the average of thederivative of the total effective energy of the system withrespect to the reaction coordinate in an ensemble whererFA is constrained to the value rÄFA. The second term isproportional to the apparent (centrifugal) force on rFA atrÄFA and is negligible in the present case.

Acknowledgments

We thank A. Honegger for the model of the ¯uor-escein-4D5Flu scFv fragment, and S. Jung and F. Schwe-singer for useful discussions. E.P. was supported by aMarie Curie fellowship of the European Union and A.C.by the Swiss National Science Foundation (grant 31-53604.98) the NCCR Structural Biology and the CancerSociety of the Canton ZuÈ rich. The work in Strasbourgwas supported by the CNRS (ESA 7006), by the Minis-teÁre de l'Education Nationale, de la Recherche et de laTechnologie and by a grant from the Association pour laRecherche contre le Cancer; that done in Harvard wassupported, in part, by a grant from NIH.

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Edited by B. Honig

(Received 3 August 2001; received in revised form 21 September 2001; accepted 21 September 2001)

Forces and Energetics of Hapten-Antibody Dissociation: A ... · Forces and Energetics of Hapten-Antibody Dissociation: A Biased Molecular Dynamics Simulation Study Emanuele Paci1,2,

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