3034
J. Phys. Chem. 1993,97, 3034-3039
Anomalous Kinetics in Antibody-Antigen Interactionst Thomas P. Theriaultt and Harden M. McConnell Department of Chemistry, Stanford University, Stanford, California 94305
Arm K. Singhal and Gordon S. Rule' Department of Biochemistry, University of Virginia, Charlottesville, Virginia 22908 Received: October I , 1992; In Final Form: January 6, 1993
The kinetics of hapten binding have been studied for one of a series of 12 anti-dinitrophenyl antibodies (Leahy; et al. Proc. Natl. Acad. Sei. U.S.A. 1988 85,3661-3665). The kinetic on rate is 100-fold less than the diffusion limited rate, Furthermore, the kinetic on rate shows an anomalous temperature dependence that indicates the presence of two or more conformations of the antibody. Proton NMR spectra of the H3,5protons of tyrosine residues show a temperature dependence of the conformation of one Tyr residue which is found in the combining site. Computer modeling of the structure of this antibody reveals that a portion of the heavy chain D region forms a loop structure which may be capable of inhibiting the access of the hapten to the combining site. Molecular dynamics calculations indicate that this loop is found in a single conformation at 5 OC,but adopts multiple conformations a t 45 O C .
Understanding of the kinetics of protein-ligand interactions is an important step in understanding the mechanism of molecular recognition. Immunoglobulins provide a rich source of systems for the study of protein-ligand interactions because of the large number of different immunoglobulins which are produced by the immune response toward a single antigen. This characteristic is particularly important as it provides a mechanism of identifying additional structural motifs for study that may not be produced by single residue alterations (e.g. by site-directed mutagenesis) of an existing motif. Once a broad repertoire of binding motifs is identified, certain members can be studied in more detail using the techniques of site-directed Of particular interest is the effect of antigen binding on the structure of the antibody. Some of the important questions to be answered are which groups on the antibody participate in binding and whether structural changes in the antibody occur during binding. There have been a number of recent crystallographic studies directed at determining the molecular events that occur during formation of an antibody-antigen complex. Most of these have been restricted to elucidating the specific molecular contacts between antibodies and haptens because of the lack of suitable crystals of the unliganded antibody.M However, in a small number of cases, it has been possible to obtain crystals of both the liganded and free antibody. These studies have shown that binding of peptide to antibody can result in either little or no structural change in the antibody7 or significant conformational changes of hypervariable loops.8 Although structural studies are clearly important in defining antigenic recognition in particular and protein-ligand interactions in general, it is necessary to correlate structure with binding kinetics and binding free energies. Toward this end, a series of 12 anti-dinitrophenyl monoclonal antibodies have been isolated and characterized for the purpose of relating the structure of the antibody to the kinetics of ligand bindingag One of these, AN02,
has been extensively characterized in terms of crystal structure,1° ligand binding kinetics,I* and magnetic resonance.12-16 Studies of this type have provided information on the structure' 1 ~ 1 2and dynamicsll of the hapten binding region of the protein. Measurements of NMR lifetime broadening of resonance lines from the unbound hapten have been used to determine the kinetic on and off rates of hapten binding to AN02." These rates display a normal temperature dependence and show that the kinetic on rate is diffusion limited. That is, the on rate constant is 7 X IO8 l/(mol.s) a t 35 OC, and the activation enthalpy for the on rate, 5.1 kcal/mol, is close to that observed for diffusion in water. Of the remaining 11 anti-dinitrophenyl antibodies, two were composed of heavy and light chains from the same subgroupI7 as AN02. The sequence homology between one of these antibodies (ANO1) and AN02, is very high with a sequence identity of 77% on the heavy chain and 88% on the light chain. Most of the differences are due to conservative replacements. However, an additional four residues are found in the D region of the heavy chain of ANOl. To investigate the effect of these extra residues on the binding kinetics of hapten to the antibody, the kinetic on rate of hapten binding to ANOl was measured as a function of temperature. The observed rate is 2 orders of magnitude slower than that observed with AN02. A computer model of ANOl indicates that these additional residues form a loop that partly occludes the binding site of the hapten. This may explain the lower intrinsic on rate. The temperature dependence of the kinetic on rate does not follow the Arrhenius equation; the on rate decreases as the temperature is increased. Molecular dynamics calculations on both ANOl and AN02 indicate that the D region loop of ANOl can assume multiple conformations at high temperatures but is found in a single conformation at low temperatures. In contrast, the same region of AN02 remains in a single conformation over the entire temperature range. Proton NMR spectra of the tyrosine residues in AN01 show evidence for conformational heterogeneity in the combining site region. This heterogeneity may explain the anomalous temperature dependence of hapten binding in AN01 .
*To whom correspondence should be addressed at Department of Biochemistry, Box 440 Jordan Hall, University of Virginia Schoolof Medicine, Charlottesville, VA 22908. Phone: (804) 924-2683. t The abbreviations used are as follows: DNP, dinitrophenyl;DNP-diGly, dinitrophenyl (bis)glycine; DNP-SL,l-(N-2,2,6,6-tetramethyIpiperidinyl-loxy-4-amino)-S-(N-(2-aminoethyl)amino)-2,4-dinitrobenzene. f Current address: Department of Chemistry, California Institute of Technology, Pasadena, CA 91 125.
Materials and Methods Sample Preparation. The generation of the cell lines producing AN01 and AN02 can be found in ref 9 and ref 18, respectively. The synthesis of the haptens (DNP-SL and DNP-diGly) can be found in ref 18. The chemical structure of these haptens can be
Introduction
0022-3654/93/2097-3034%04.00/0
0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 3035
Antibody-Antigen Interactions found in ref 11. Growth conditions for the hybridoma cells in deuterated media have been described previo~s1y.l~H3,5-Tyr labeled immunoglobulins were obtained by substituting L-tryp-
tophan-2',4',5',6',l-d5, ~-(4-hydroxypheny1-2,6-d2)alanine-2-d1 (tyrosine), and ~-(phenyl-ds)alanine-3,3-d2for Trp, Tyr, and Phe in the culture media. Fab fragments were prepared from purified immunoglobulin by limited proteolysis.12 NMR Spectroscopy and Paramagnetic Broadening. NMR spectra were obtained on a General Electric GN-series spectrometer operating at 500 MHz (proton). Approximately 6000 scans of 16K data points were obtained. The spectral width was 8000 Hz, giving a digital resolution of 0.5 Hz/point. Nuclear Overhauser effects (NOE) were detected by saturating the Tyr resonance lines with a selective pulse for 150 ms. Irradiation at a region devoid of resonances was used for the control. Residual water protons were presaturated, and chemical shifts are given relative to a tetramethylsilane standard. Paramagnetic broadening due to a nitroxide label hapten was used to estimate the distance between the nitroxide moiety and the Tyr residues (see ref 13). Binding of the hapten to ANOl produces relatively small changes in the chemical shift of the Tyr resonance lines (see Table 11). Since the exchange kinetics are faster than these chemical shift differences, the contribution of lifetime broadening to the line width of the Tyr residues will be small. The rate of chemical exchange is also faster than the spin-spin relaxation rate, except for residues that are less than 8 A from the nitroxide, thus to a good approximation the T2 can be considered to be the average between bound and free states. If the rate of paramagnetic relaxation is faster than the exchange rate, then the slow exchange limit applies and the observed line width is related to the lifetime of the free antibody. Under conditions of fast exchange of the paramagnetic hapten, the effect of the averaged T2 on the peak amplitude of a resonance line is given by eq 1. Where T2, is the diamagnetic contribution to the
spin-spin relaxation, TzPis the paramagnetic contribution to the spin-spin relaxation, and f is the fractional occupancy of the combining site by the nitroxide labeled hapten. The distance between the proton and the nitroxide can be calculated from eq 2. A difference spectrum at an occupancy f is calculated by
subtracting a spectrum obtained when the fractional occupancy of the binding site by DNP-SL is f from a spectrum obtained in the absence of DNP-SL. The intensity of the line in a difference spectrum (Z(0) - 1 0 ) is given by eq 3. The amplitude of the resonance lines in difference spectra obtained at different occupancies (f) of DNP-SL were fitted to eq 3 using a nonlinear least-squares program NONLIN19 in order to obtain TzP. fT2:
Zy) =-
a
1
T2p + f T 2 "
(3)
Binding Constants and Chemical Kinetics. The affinity constants of DNP-diGly for ANOl in the temperature range of 1545 OC were obtained by quantifying the chemical shift of Tyr resonance "E" (see Figure 3A,B) as a function of the DNP-diGly concentration. The affinity constant at 5 OC was obtained using fluorescence quenching. The kinetic on rates were obtained by measuring the line width of an aromatic proton on the unbound hapten as described in ref 11. Note that the calculation of the on rate assumes that the exchange rate of the hapten is slow compared to the chemical shift difference between the resonance lines that arise from bound and free hapten. On the basis of the observed shift of the resonance line from the unbound DNP diGly, as additional ANOl is added to the sample, the unbound hapten
TABLE I: Equilibrium Constants and On Rates for AN01 and AN02 ANOl temp ("C) 45 37c 25 15 5
AN02"
Kcs?(M-l)
1pkm (M-l s-')
l p K , (M-I)
1O-*k,, (M-1 s-I)
640 f 190 1660 130 2400 f 300 4500 f 600 12800 f 1300
1.0 f 0.1 3.3 0.2 4.6 f 0.4 5.0 f 0.6 6.9 f 0.8
0.8 1.7 3.8 8.0 16.0
7.6 7.1 5.0 3.5 2.6
*
*
*
a Values obtained from ref 11. The error in Kcs was obtained from least-squares fit of binding data. 35 OC for measurements on AN02.
is in slow exchange with the bound hapten. Thus the kinetic on rate can be obtained from the line width of the unbound hapten. Molecular Modeling and Dynamics. Molecular modeling was performed using Sybyl Molecular Modeling Software (version 6.0, Tripos Associates, St Louis, MO) implemented on either a Silicon Graphics 4D/35 or Personal Iris platform. All calculations are based on the Kollman United force field parameter set.20-21 The empirical energy functions are as described for the CHARMM program.22 The starting structure for ANOl was based on a model of ANOl provided by M. Levitt (Stanford University). The starting structure for AN02 was based on the crystal structureof AN02.I0 These structures were subject to a maximum of 40cycles of simplex minimization followed by 160 cycles of steepest descent minimization. Refinement was terminated when the root mean square gradient in energy between successive iterations of minimization was less than or equal to 1.0 kcal/(mol.A2). This structure was then equilibrated to the temperature of interest in a manner similar to that described by Brooks et a1.22 Briefly, this involved 0.1 ps of molecular dynamics at 10 K intervals beginning at 5 K. During these cycles the starting velocities were randomly assigned on the basisof an appropriateGaussian distribution for that temperature. Between each cycle of molecular dynamics the structure was minimized by 100 cycles of steepest descent minimization. When the temperatureof interest (5,45, or 85 "C) was reached, a single cycle of 10.0-ps dynamics was performed with reassignment of random velocities every 0.5 ps and minimization of the structure every 2 ps by 200 cycles of steepest descent minimization with a 1.O kcal/(mol.A2) termination option. The resultant structure was used to begin 25.0 (AN02) or 50.0 ps (ANO1) of molecular dynamics at each temperature. The Shake23option was enabled during this entire time to constrain bond lengths.
Results and Discussion The binding of hapten to antibody consists of at least two steps:24 an encounter step, followed by the actual binding step. The rate of the encounter step is regulated by intermolecular collisions, while the rate of the binding step depends on the rate of formation of specific molecular interactions between the hapten and the antibody. If the observed kinetic on rate is comparable to the collisional rate, then it is reasonable to assume that the binding step does not involve an extensive conformational change of the antibody. If the kinetic on rate is slower than the collisional rate, then the formation of the collisional complex is not rate limiting. Instead there is a significant activation energy of the binding step. This activation barrier may include desolvation and rearrangement of the conformation of the protein. The equilibrium constants and the kinetic on rates for DNPdiGly binding to ANOl and AN02, measured at various temperatures, are presented in Table I. A plot of In k,, versus 1/ T is shown in Figure 1. The on rate of DNP-diGly to AN02 increases with temperature, and the temperature dependence indicates an activation energy of 5.1 kcal/mol for binding. Since the kinetic on rate is similar to the collisional rate, and the activation enthalpy for binding is similar to that for free diffusion, the binding pocket in AN02 is pre-formed.
3036
The Journal of Physical Chemistry, Vol. 97, No. 12, i'993
I 20
3.1
3.2
3.4
3.3
I/T
x
~
3.5 O
~
3.6 K
O
Figure 1. Kinetic on rates of DNP-diGly as a function of 1 /Tfor ANOl (open symbols) and AN02 (closed symbols).
In contrast, the kinetic rates for DNP-diGly binding to ANOl are much slower than those observed for AN02; thus the rate of hapten binding to ANOl is regulated by the actual binding step. In addition to a reduced on rate, the on rate of hapten binding to ANOl also decreases with temperature. This indicates that the system must exist in two (or more) different conformational states. For the purposes of discussion, a simplifying model which assumes two states will be developed as an aid to understanding the system in thermodynamics terms. State A is more heavily populated at low temperature, while state B is increasingly populated at higher temperatures. In this model, the kinetic on rate of the low temperature A state is faster than that of the high temperature B state. The temperature dependence of the kinetic on rate is given in eq 4 (see Appendix for derivation). In eq 4,
k,, is the observed on rate, k t and k;, are the kinetic on rates for the two states, AHABis the equilibrium enthalpy difference between the two states, AH: and AH; are the activation enthalpies for the kinetic on rates of each state, and KABis the equilibrium constant for the conversion of A to B. The temperature dependence of the on rate is complex and depends on the relative values of the equilibrium constant, the equilibrium enthalpy, and the activation enthalpies (see Appendix for a detailed discussion). In the case of DNP-diGly binding to ANOl, the observed change in k,, is negative over the entire temperature range investigated. In order for this to occur AHAB must be positive since the term containing AHAB is the only negative term in eq 4. Since AGABdecreases as the temperature is raised, it follows that the entropy difference between A and B (ASAB) is also positive. The difference in the temperature dependence of binding for ANOl versus AN02 must be a result of structural differences between the two antibodies. The sequences of the complementarity-determining regions (CDR) for these antibodies are given in ref 4. The most noticeable difference between these two sequences is the presence of an additional four residues in CDR3
Theriault et al. of the y chain (CDR3y) in ANO1; all other CDRs are the same length. The sequence of CDR3y in ANOl is: CAREDDGYYIFDYWC. The corresponding sequence in AN02 is: CARGWPLAYWC (bold typeface indicates alignment of homologous residues). Figure 2A shows a model of ANO1, and Figure 2B shows the crystal structure of AN02. In AN02, the binding site for hapten is found between Trp91L and Trp96H. In ANOl a binding site for the hapten is presumed to be between Trp9 1L and CDR3y. The additional residues in CDR3y in ANOl do not affect the overall structure of the other CDRs in the combining site. Thus the major structural difference between ANOl and AN02 is located in CDR3y. It is apparent from Figure 2A that the additional four residues in CDR3y of ANOl protrude into the hapten binding pocket of ANOl and conformational rearrangements of the antibody will have to occur for the hapten to bind. To gain some insight into the differences between the high and low temperature states of ANOl ,molecular dynamics calculations were performed at three different temperatures for both ANOl and AN02. The two lowest temperatures (5 and 45 "C) were chosen because they correspond to the limits of the temperature range over which the binding was measured. Calculations were also performed at a higher temperature (85 "C) to take into account differencesbetween the actual experimental temperatures and the temperature of thedynamics simulations. Representative structures from the calculations are shown in Figure 2C-H. These calculations indicate that CDR3y in AN02 (Figure 2D, F, H) does not undergo any significant conformational changes over the temperature range studied. Some large amplitude motions are seen at 85 OC, but these do not result in a change in the conformation of CDR3y. Specifically, the relative orientation of Trp91L and Trp96H is not altered. In contrast, the conformation of the CDR3y region of ANOl changes with temperatures. At 5 OC this region remains in a single conformer during the entire 50 ps of molecular dynamics (Figure 2C). At 45 OC (Figure 2E) this loop assumes three different conformational states during 50 ps of molecular dynamics. The first state, which is different than that observed at 5 OC, persists for the first 15 ps of dynamics (colored yellow in Figure 2E). This state is then replaced by a conformation that is similar to that observed at 5 OC (not shown). This second state persists for 10 ps before being replaced by a third state, which is also different from that observed at 5 OC (colored green). This third state persists for 15 ps before the molecule returns to the second, 5 OC-like state, where it remains for the last 10 ps of the simulation. Note that the motion of Trp91L is similar to that seen for AN02 at the same temperature. At 85 OC CDR3y has assumed a somewhat disordered conformation which has some characteristics that are similar to the first state at 45 OC. Differences between the two states include a re-orientation of TyrlOOH and Trp91L. Although there are large amplitude motions of the atoms in CDR3y and changes in the torsional angles during the simulation of 85 OC, the overall conformation of CDR3y is the same throughout the entire 50 ps of molecular dynamics, indicating that this ensemble of structures has a moderately long lifetime. The molecular dynamics calculations show a clear difference between the effect of temperature on the conformation of ANOl and AN02. Experimental evidence for such heterogeneity can be observed in the NMR spectrum of the H3,S-protons of Tyr residues in ANOl. Proton NMR difference spectra, obtained at 37 OC, are shown in Figure 3. These spectra show signals from the 12TyrresiduesinANOl whicharewithin 18Aofthenitroxide group. Resonance F shows a change in line shape as a result of hapten binding. At 37 OC and in the absence of hapten (Figure 3A) it is found as a doublet, while in the presence of hapten it is a single resonance line. The effect of hapten binding on the line shapeof resonance F is also dependent on temperature. Figure 4 shows NMR spectra obtained in the absence and presence of haptenat25 and45 "C.Theeffectofhapten bindingonresonance
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 3037
Antibody-Antigen Interactions
I
I
1
I Figure 2. Molecular structure and dynamics calculations on ANOl and AN02. The main chain atoms and some selected side chain atoms are shown for the Fv portion of ANOl (left panels: A, C, E, G, top to bottom) and AN02 (right panels; B, D, F, H, top to bottom). The light chain is located on the top right and the heavy chain is located on the bottom left of each panel. In all of the panels the backbone tracing is shown in blue. Panels A and B show the hapten (DNP-SL, colored cyan) docked in the combining site. Trp91L and CDR3y are shown in red for both ANOl and AN02. Tyr residues that are less than 18 A from the nitroxide are colored violet. The remaining six panels show molecular dynamics calculations at 5 OC (panels C and D), 45 OC (panels E and F), and at 85 OC (panels G and H). In all of these six panels the red structure is that obtained after 5 ps of molecular dynamics at 5 OC. ANOl (Panels C, E, G): In panel C structures obtained after 25 ps (yellow) and 50 ps (green) of dynamics at 5 OC are shown. In panel E the two conformers that are different from the 5 OC structure are shown in yellow ( 5 ps at 45 "C) and green (25 ps at 45 "C). Panel G shows structures calculated after 5 ps (yellow) and 50 ps (green) of molecular dynamics at 85 "C.The orientation of CDR3y in these panels is such that TyrlOOH is to the right. AN02 (Panels D, F, H): In panel D the structure obtained after 25 ps of molecular dynamics at 5 OC is shown in yellow. In panel F structures obtained after 5 ps (yellow) and 25 ps (green) of molecular dynamics at 45 OC are shown. Panel H is the same as panel F, except that the calculations were performed at 85 OC.
3038 The Journal of Physical Chemistry, Vol. 97, No. 12, 199'3 D
1.-
7.5
6.5
7.0
C
6.0
S (ppm) Figure 3. Proton NMR spectra of H3,5-Tyr ANO1. All spectra are differencespectra that are calculated by subtracting the spectra obtained
with the Fab saturated with DNB-SL from the spectra obtained under the indicated conditions. Spectrum A shows the aromatic region of the NMR spectrum in the absence of hapten. Spectrum B shows the effect of saturating levels of DNP-diGly. Spectra C and D show the effect of adding DNP-SL to the sample in spectrum B to give 5.7% occupancy (C) and 1.5%occupancy (D). The proteinconcentration in these spectra was 124 pM. The concentration of DNP-diGly was 2.4 mM in spectra E D . The temperature of the sample was 37 OC, and 5000 scans were accumulated for each spectra.
I I
7.0
II
25°C
7.0
6.0
45oc
6.0
8 (ppm)
F i p e 4 . Effect of temperature on the proton NMR spectra of H3,S-Tyr ANOl, These spectra are not difference spectra. Spectra were accumulated in the absence of DNP-diGly at 25 OC (bottom left) or 45 OC (bottom right) or in the presence of 5.4 mM DNP-diGly at 25 OC (top left) or at 45 OC (top right). An arrow indicates the position of resonance F in each spectrum. The protein concentration was 120 pM.
F is similar at 45 OC as that observed at 37 OC. In contrast, at 25 OC hapten binding does not cause significant changes of this resonance line. Although the above NMR data support the existence of thermally induced conformational heterogeneity in ANOl, it is necessary to determine if the resonance involved (F) is likely to arise from a Tyr residue in CDR3y (either Tyr99H or Tyrl OOH). By varying the concentration of DNP-SL (Figure 3C,D), it is possible to obtain quantitative informationon the distance between the hapten and the Tyr residues. This distance information is presented in Table 11, along with the chemical shifts and the changes in chemical shift due to binding of DNP-diGly. Assuming that the general location of the DNP ring is the same in both antibodies, it is possible to use the crystallographic structure of the complex between AN02 and DNP-SL to obtain a number of tentative resonance assignments for the Tyr resonances in the spectrum of ANO1. In AN02 the hapten is found between the
Theriault et al.
TABLE II: Properties of Tyr Resonances in ANOP resonance 6 (ppm) Ah (ppm) R (A) assignment (db(A)) A 7.58 -0.04 16 B 7.07 0.00 13 C 6.84 0.00 9 Tyr49L(10) D 6.78 0.00 15 E 6.76 +0.17 9 TyrlOOH (9) F 6.73,6.75 +0.02 10 Tyr99H (11) G 6.63 0.00 15 H 6.63 -0.03 9 Tyr'llL(9) I 6.59 -0.04 k,: and neglect the terms in k,", in eq A8, we have a maximum in ko, when
or
That is, at lower temperatures& is smaller and dk,,/dTis positive, and at higher temperatures, fB is larger and dk,,/dTis negative. Note that this maximum in k,, requires the interplay of KAB, AHAB,and AH: together with the supposition that ko", >> k.:, The latter implies AHAB> 0 in order for dk,,/dT to attain values less than zero.
References and Notes (1) Glockshuber, R.; Stadlmuler, J.; Pluckthun, A. Biochemistry 1991,
30, 3049.
McManus, S.; Riechman, L. Biochemistry 1991, 30, 5851. Yamout, M.; McConnell, H. M. Results to be published. Cygler, M.; Rose, D. R.; Bundle, D. R. Science 1991, 253, 442. Herron, J. N.; He, X.M.; Mason, M. L.; Voss, E. W.; Edmundson, A. B. Proteins 1989, 5, 271. (6) Satow, Y.; Cohen, G. H.; Padlan, E. A.; Davies, D. R. J. Mol. Biol. 1986, 190, 593. (7) Stanfield, R. L.; Fieser, T. M.; Lerner, R. A.; Wilson, I. A. Science 1990, 248, 713. (8) Rini, J. M.; Schulze-Gahmen, U.; Wilson, I. A. Science 1992, 255, (2) (3) (4) (5)
959. (9) Leahy, D. J.; Rule, G.S.; Whittaker, M. M.;McConnell, H. M. h o c . Natl. Acad. Sci. U.S.A. 1988, 85, 3661. (10) Brunger, A. T.; Leahy, D. J.; Hynes, T. R.; Fox, R. 0. J . Mol. Biol. 1991, 221, 239. (11) Theriault, T. P.; Leahy, D. J.; Levitt, M.; McConnell, H. M.;Rule, G. S. J. Mol. Biol. 1991, 221, 257. (12) Anglister, J.; Frey, T.; McConnell, H. M. Biochemistry 1984, 23, 1138. (1 3) Anglister, J.; Frey, T.; McConnell, H. M. Biochemistry 1984, 23, 5372. (14) Frey, T.; Anglister, J.; McConnell, H. M. Biochemistry 1984, 23, 6470. (15) Anglister, J.; Frey, T.; McConnell, H. M. Nature 1985, 315, 65. (16) Anglister, J.; Bond, M. W.; Frey, T.; Leahy, D. J.; Levitt, M.; McConnell, H. M.; Rule, G. S.; Tomasello, J.; Whittaker, M. M. Biochemistry 1987, 26, 6058. (17) Kabat, E. A.; Wu, T. T.; Bilofsky, H.;Redi-Miller, M.; Perry, H.
Sequences of Proteins of Immunological Interest; National Institutes of
Health: Bethesda, MD, 1983. (18) Balakrishnan, I(.; Hsu, F. J.; Hafeman, D. G.; McConnell, H.M. Biochim. Biophys. Acta 1982, 721 30. (19) Johnson, M. L.; Frasier, S. G. Methods Enzymol. 1985, 117, 301. (20) Weiner, S. J.; Kollman, P. A,; Case, D. A.; Singh, U. C.; Ghio, C.; Alagona, G.; Profeta, S.; Weiner, P. K. J. Am. Chem. SOC.1984, 106, 765. (21) Weiner,S. J.;Kollman,P.A.;Nguyen,D.T.; Case,D.A. J. Comput. Chem. 1986, 7 , 230. (22) Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.; Swammathan, S.; Karplus, M. J. Comput. Chem. 1983, 4, 187. (23) van Gunsteren, W. F.;Berendsen, H. J. C. Mol. Phys. 1977, 34, ~
dfA dkon -=(k,%-kB)-+f On d T dT
dktn
A
-+f dT
dktn
dT
(A4)
1311. (24) Pecht, I.; Lancet, D. In Chemical Relaxation in Molecular Biology; Pecht, I., Rigler, R., Eds.; Springer-Verlag: Berlin, New York, 1977.
