J. Phys. Chem. 1982, 86,2314-2321
2314
is outermost, and the geminal methyne proton is closer to the plane. Certainly, this structure is also acceptable energetically since its geometry is considered to be the most suitable one among those which maintain an effective CT-attractive interaction between both aromatic planes of 7TH and TAPA. This structure also minimizes an electronic repulsive interaction between the propionic chain of TAPA and the 7r-electron cloud of 7TH. The small discrimination energies (loc. cit.) are understood by a resemblance of both diastereomersin this basic geometry. On the other hand, a structural difference can be recognized in terms of the ratio AcR Acs where a deviation from unity provides a measure3 of the structural discrepancy. Table I1 indicates that all of the values of A C R / k Sfor the ring protons of 7TH and TAPA are near to unity (the mean value being 1.21 in a range of 0.84-1.45) while the values for protons in the propionic group are far from unity. Thus, a major difference in the geometries of the two diastereomers should exist in the arrangement of the propionic group of TAPA toward the staggered plane of 7TH. Judging from the large AcR/AcS values in the propionic group, one can guess that the chiral moiety in
4
the (R)TAPA component is located close to the upward winding terminal of (P)7TH, while that in the (S)TAPA component is kept over the upper side of the downward terminal. Such a change in the location of the chiral moiety from one diastereomer to the other may cause a slight change in the overlapping mode between both aromatic planes of TAPA and 7TH, thus leading to the s m a l l up and down shifts about unity in the AcR/ Acs values for the ring protons. On the basis of the above discussion, we can illustrate the two diastereomeric pairs (shown in Figure 3) where a different feature in the allocation of the chiral moiety is stressed. The spatial orientation thus inferred correlates well with the previously determined energetics: the (P)7TH-(R)TAPA pair may have a tendency to be destabilized by an electronic repulsion between the propionic acid moiety of TAPA and the (P)7TH terminal, while the (P)7TH-(S)TAPA pair may avoid such a repulsion and be stabilized through a net charge-transfer interaction. In conclusion, the stability difference in the present diastereomeric pairs is confirmed consistently from both thermodynamic and structural parameters.
Rate Theory for Gated Diffusion-Influenced Ligand Binding to Proteins Scott H. Northrup,’ Fahlmeh Zarrin,+ Department of Chemistry, Tennessee TechnologicalUniversity, Cmkeville, Tennessee 3850 1
and J. Andrew McCammon’ mpartment of Chemistry, University of Houston, Houston, Texas 77004 (Received: November 18, 1981; In Final Form: February 4, 1982)
The rate of binding of ligands to proteins may be determined not only by the relative diffusion rate of species through the solvent medium but also by the accessibility of the binding site. Because of the inherent flexibility and internal motion of proteins, this accessibility may fluctuate on the time scale of reaction, thereby causing the intrinsic reactivity of the protein to be a time-dependentquantity. Here, we present a general formulation of the kinetics of such gated reactions. Approximate analytical expressions for the rate constant are obtained for important limiting cases. These compare favorably with exact numerically obtained values for the gated reaction rate constant over a wide range of system parameters.
species due to stereochemical orientation or other static I. Introduction geometric features of binding sites.’G17 The binding of ligands to proteins often occurs sufficiently rapidly that the rate of diffusional transport of species can influence the overall rate.’ Much attention (1) C. R. Cantor and P. R. Schimmel, “Biophysical Chemistry”. Part has been focused on these so-called diffusion-influenced 111, ‘The Behavior of Biological Macromolecules”, W. H. Freeman, San reactions.2 Since the early theory of Smoluch~wski,~ Francisco, 1980. (2)R.M. Noyes in ‘Progress in Reaction Kinetics”, Vol. 1, Pergamon, generalizations have been made to include a number of New York, 1961, Chapter 5. important and interesting features: the influence of (3)M. v. Smoluchowski, Phys. Z.,17,557 (1916). (4)S.H. Northrup and J. T. Hynes, J. Chem. Phys., 68,3203 (1978). short-range activation barriers on the intrinsic reactivity (5)S.H. Northrup and J. T. Hynes, J. Chem. Phys., 71,871 (1979). of juxtaposed particle^;^ the modulation of encounter rates (6)P. Debye, Trans. Electrochem. SOC.,82,265 (1943). due to interparticle potentials extending beyond the col(7) J. M. Deutch and B. U. Felderhof, J. Chem. Phys., 59,1669(1973). (8) P. G. Wolynes and J. A. McCammon, Macromolecules, 10, 86 lision separation (arising from solvent structural effects5 (1977). and/or electrostatic interactions6);short-range modulation (9)R. Samson and J. M. Deutch, J . Chem. Phys., 67,847 (1977). of the diffusion rate due to hydrodynamic interaction^;^^^,^ (10)H. Sano, J . Chem. Phys., 74, 1394 (1981). effects of competition between reactants for a depleted (11)K. Solc and W. H. Stockmayer, J . Chem. Phys., 54,2981 (1971). (12)W. Scheider, J. Phys. Chem., 76,349 (1972). population of coreactant~;~J~ and anisotropic reactivity of ‘Department of Chemistry, University of Wyoming, Laramie, WY.
(13)K. S.Schmitz and J. M. Schurr, J . Phys. Chem., 76,534 (1972). (14)M. Doi, Chem. Phys., 11, 115 (1975). (15)T.L. Hill, Proc. Nut[. Acad. Sci. U.S.A., 72,4918 (1975). (16)R. Samson and J. M. Deutch, J . Chem. Phys., 68, 285 (1978).
0022-3654/82/2086-23 14$01.25/0 0 1982 American Chemical Society
Protein-Ligand Reaction Rate Constants
In this paperla we treat another complexity that may arise particularly with regard to the binding of ligands to proteins. Proteins and other complicated chemical entities may fluctuate among a number of internal states affording different binding site a c c e ~ s i b i l i t y . ~The ~ ~ ~ internal ~ motion of the protein (e.g., the hinge-bending motionz1of lysozyme19)may dynamically alter its intrinsic reactivity on the time scale of diffusional relaxation. We refer to this kind of reaction as a “gated” reaction, since the protein may be thought to possess an opening and closing gate controlling binding site reactivity. Our major goal will be to derive a bimolecular rate constant for gated reactions which accounts for the influence of both diffusional transport and gate dynamics on the overall reaction dynamics. Our emphasis will be primarily on the coupling between the diffusion problem and gating and to a lesser extent on the specific details of the protein gate dynamics. We assume a simple model for the latter and include some brief remarks on more general gate models in our conclusion. The outline of the paper is as follows. In section I1 we describe a simple working model of a gated diffusion-influenced reaction. There we show in a general way how the bimolecular rate constant is related to microscopic quantities such as the reactant spatial distribution and the time-varying intrinsic reactivity. In section I11 we show formally via Green’s functions how the diffusion equation modified to include gating is solved for the spatial distribution. Because of the analytical intractability of the formal result, we develop in section IV an approximate analytical expression for the rate constant based on an assumption valid in important cases. We discuss the domain of validity of this expression and its behavior in two interesting limiting cases. In section V we compare the convenient approximate rate constant with more cumbersome exact numerical solutions for a wide variety of gating conditions. We summarize and offer some concluding remarks in section VI. 11. General Formulation Our working model of a gated reaction system can be described as follows. The potentially reactive particles (e.g., a protein (P) and ligand (L)) are considered to be spherical molecules immersed in a viscous medium. They exhibit negligible interactions, hydrodynamic or otherwise, except for excluded volume interactions at separation distance r less than contact separation R. Furthermore, they can undergo a chemical reaction regardless of stereochemical orientation at contact separation. The protein reactivity is modulated by an internal gate that may be in either an open (reactive) or closed (nonreactive) state. The assumption of isotropic reactivity is perhaps the most unrealistic restriction of the model as it applies to ligand binding to proteins. However, as we wish to focus on the effects of the protein gate, we exclude anisotropic reactivity effects to avoid the additional complexities their treatment would entail. The bimolecular rate law which would be observed in a typical kinetics experiment is22923
(17)K. Chou, T.Li, and S. Forsen, Biophys. Chem., 12,265 (1980). (18)J. A. McCammon and S. H. Northrup,Nature (London),293,316 (1981). (19)J. A. McCammon, B. R. Gelin, M. Karplus, and P.G. Wolynes, Nature (London),262,352 (1976). (20)J. A. McCammon and P. G. Wolynes, J. Chem. Phys., 66,1452 (1977). (21)0.B. Ptitsyn, FEBS Lett., 93, 1 (1978).
The Journal of Physical Chemistty, Vol. 86, No. 13, 1982 2315
Here Cp(t) is the concentration of unliganded protein (assumed to be infinitely dilute), CLois the concentration of ligand (assumed to be present in excess), and k is the rate constant. Under the condition of infinitely dilute protein concentration, the problem of competition between unliganded proteins for available ligands is avoided. The focus of our description of the kinetics of ligand binding to proteins is the bimolecular rate constant k. In theoretical terms k is obtained by considering the average reaction rate in an infinite ensemble of isolated proteins surrounded by ligandsaZ4The protein in each ensemble member may have an arbitrary and unique history of gate dynamics. However, it is more convenient for our purposes to consider a smaller (but still infinite) ensemble where the protein gate history of every member is identical. That is, the protein internal gates of all ensemble members are opening and closing simultaneously. This allows us to speak of a unique characteristic function h ( t )which describes the state of the protein gate for all ensemble elements. The gate function h(t)= 1 at times when the gate is open, and h(t) = 0 otherwise. Since the spatial arrangement in each ensemble member is unique, we define a spatial probability distribution function p(r,t) of unreacted protein-ligand pair separations r at time t. This function is more easily obtainable than the distribution for the larger ensemble because it describes an ensemble in which every member has been subject to the same gate dynamical history h(t). With the above ensemble definition, the rate constant is given by a time average rather than a strict ensemble average as
k = (~(t= ) )(4aR212,h(t)p(R,t))
(2.2)
Here ~ ( t is) the time-dependent reaction rate coefficient,22*23@ which is the product of the following quantities: the specific rate constant k, for reaction of pairs attaining a point on the reactive surface r = R; the area 4aR2 of the reactive surface; the gate function h(t);and the unreacted pair density p(R,t) at a point on the contact surface r = R. The density is normalized such that p(r,t) = 1 in a system in which the reactive gate has been closed for a long time period and spatial equilibrium prevails. The function ~ ( tfluctuates ) between zero (when the gate is closed) and various finite values which depend on the time-varying population of juxtaposed unreacted pairs. The time average (...) is over times sufficiently long to include a statistically large sample of open and closed gate periods and to exclude initial condition effects in the pair distribution function. It should be noted that ~ ( defined t) in eq 2.2 could not be observed in a conventional kinetics experiment because it corresponds to a special ensemble in which all gates are in phase. To observe ~ ( tone ) would require a special experiment in which all protein gates are kept in place by the action of an external influence. For the purposes of obtaining k,however, the time average of ~ ( in t )the special ensemble is formally equivalent to the full ensemble average in the steady state. This equivalence is due to the fact that the gate phase becomes unimportant when performing time averages over a large number of gate cycles. We have isolated k ip terms of two important dynamical quantities p(R,t) and h ( t ) ,which we now discuss. The (22)S. H. Northrup and J. T. Hynes, Chem. Phys. Lett., 54, 244 (1978). ~ - - _,_ .
(23)T.L. Nemzek and W. R. Ware, J.Chem. Phys., 62,477 (1975). (24)J. Yguerabide, J. Chem. Phys., 47,3049 (1967). (25)J. C.Andre, M. Bouchy, and W. R. Ware, Chem. Phys., 37, 103 (1979).
2318
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982
density p(r,t) is governed by a modified version of the well-known Smoluchowski equation for radial diffusion5 ap(r,t)/at =
rZ2( ar rzD):
- k,h(t)p6(r
- R)
(r 2 R ) (2.3)
Equation 2.3 is supplemented with the reflecting wall boundary condition a t r = R to account for the excluded volume effect. An outer boundary condition p ( a , t ) = 1 accounts for an unperturbed spatial equilibrium at large distances from the reactive surface. The sink term -k,h(t)pd(r - R ) describes irreversible loss of unreacted pairs a t r = R due to reaction.26 The function 6(r - R ) is the 6 function. Note that in a gate-closed state the sink term vanishes and eq 2.3 becomes the usual homogeneous equation for radial diffusion in the absence of reaction. The solution of eq 2.3 will be discussed in the following section. The characteristic gate function h(t)that we are considering has the form of a nonperiodic square wave of unit amplitude m
h ( t )=
c
i=1,3,5...
[O(t - ti) - O(t - ti+1)]
(2.4)
Here, O(t) is the unit step function, ti = t,, t,, t 5 , ... is an ordered set of gate opening times, and ti = t2, t4,t6, ... are gate closing times. There are certainly other possible forms of h(t),such as cases in which the protein reactivity may vary in time over a continuous range of values. Since our major emphasis in this study is the coupling between diffusion dynamics and gate dynamics, we will not be concerned with deriving h ( t ) from a model of gate dynamics. We will simply assume h ( t ) is some known function and proceed from there.
111. Formal Solution for the Gated Reaction Rate Constant The rate constant for the gated reaction described in section I1 can be obtained by solving eq 2.3 for p(R,t)using an appropriate gate function h(t)and then performing an average according to eq 2.2. The solution of eq 2.3 can be constructed formally by a convolution of known Green’s functionsz7for two simpler problems. First, relaxation of the system while in the gate-open state is described by the reactive Green’s function Grx(r,r’,t),which satisfies the same equation
k,C,,(R,r’,t)G(r - R )
(r I R ) (3.1)
The initial condition for eq 3.1 is Grx(r,r’,O)= 6(r - r?. Again, the reflecting wall boundary condition applies a t r = R , while the outer boundary condition is G,(m,r’,t) = 0. The function G,(r,r’,t) is specified in Appendix A. By Green’s theorem G,, can be used to construct the distribution p(r,t) in a system in which the gate opens at t = t, with initial distribution p(r,tl) and remains open until time t: p(r,t) = l m d r ,4~r1~G,,(r,rl,t - tl)p(rl,h) R
(3.2)
If the distribution at the gate-opening time is the equilibrium distribution, p(r,tl) = 1,the integral in eq 3.2 can be evaluated to yield an important result for the approx(26) G. Wilemski and M. Fixman, J. Chem. Phys., 58, 4009 (1973).
Northrup et al.
imate treatment in section IV. The result is also specified in Appendix A. During gate-closed periods, relaxation is described by the reflecting wall Green’s function Gmfl(r,r’,t),which obeys eq 3.1 without the source term and with the same initial and boundary conditions. The distribution at any time during a gate-closed period can be related to the distribution at the beginning of that period by eq 3.2 except with Greflreplacing G,. Grefl(r,r’,t)is specified in Appendix A. We now apply G,, and Greflto obtain the distribution function of the gated system whose characteristic gate function h ( t ) is given by eq 2.4. Suppose p(r,tl) is the distribution a t time t = t , at which the gate first opens. At any later time during the first open cycle, the distribution is given by eq 3.2. During the first closed cycle, which begins at t = t z , the distribution would then be
Successivealternate convolution of G, and GEflwould then yield the distribution a t a time during any desired subsequent reactive or unreactive cycle. For example, at time t during the reactive period t,+l > t > t, (where n is odd) the distribution is given formally by p(r,t) = (47r)”$dr, rn2Grx(r,rn,t - t,) x
Here, CY = rx (refl) when i is odd (even). Equation 3.4 clearly illustrates the complexity involved in obtaining an exact analytic solution to the general gate problem. A more convenient alternative to the analytical approach is the use of various numerical methods. In this study we will employ the well-known finite difference methodz8for solving eq 2.3 when an exact solution is desired. A brief description of the application of this method to gated reactions is given in Appendix B. Under certain limiting conditions approximate analytical solutions for p(r,t) based on eq 3.2 can be employed to obtain quite accurate values for k. This is the subject of the next section.
IV. Approximate Analytical Theories for the Rate Constant In this section we will discuss certain limiting conditions in which analytical expressions can be obtained for gated diffusion-influenced reaction rate constants. However, let us first review the simpler case of an ungated diffusioninfluenced reaction. This case is considered not only for reference but because the result serves as a basis for an approximate treatment of certain gated cases. In the ungated case the gate function h(t) = 1 for all times t > 0 in a system initially in spatial (but not chemical) equilibrium. The solution of eq 2.3 with h ( t )= 1 is given by eq 3.2 with p(r’,O) = 1. The explicit mathematical expression for p(r,t) is given by eq A.4 in Appendix A. The timedependent rate coefficient for the ungated case22~ “ ( is t) then simply related to the contact distribution function as K,(t)
= 4TR2k,p(R,t)
(4.1)
(27) P. M. Morse and H. Feshbach, ‘Methods of Theoretical Physics”, Vol. 1, McGraw-Hill, New York, 1953, Chapter 7. (28) See, e.g., G. Dahlquist and A. Bjorck, “Numerical Methods”, translated by N. Anderson, Prentice-Hall, Englewood Cliffs, NJ, 1974, p 383.
The Journal of Physical chemistry, Vol. 86,No. 13, 1982 2317
Protein-Ligand Reaction Rate Constants
Equation 4.7, which is the fundamental result of this section, has the following simple physical interpretation. p(R,t) = 1 - y(1 + y)-'[l - eoZterfc ( p t 1 / 2 ) ] (4.2) The rate constant k is equal to the fraction of time the gate ) is open, (h(t )), times the average rate constant K ( T ~for Here, 7d-1/2(1 y), y = Rk,/D, Td = R2/D, and erfc ligand binding during any open peri.od rowhich has started is the complementary error function. In physical terms, with an equilibrium spatial distribution. We now discuss y is a dimensionless coupling constant which gauges the the limits of validity of eq 4.7. extent to which the reaction is diffusion ~ontrolled.~ For As stated previously, the result k = K ( T (~h) )is valid only y >> 1, the intrinsic reactive step proceeds much more for gated reactions in which the characteristic gate-closed rapidly than the diffusion step, and the rate of diffusion time 7,is sufficiently long to allow the spatial distribution controls the overall reaction rate. For y t > ti, odd i) =0 (ti+2 > t > ti+', odd i) (4.5) Equation 4.10 holds when all but the first term vanish, or Here, all times ti where i is odd are gate-opening times, when 7o = ti+' - ti (odd i) and 7, = ti+' - ti (even i). Since K ( t ) is a period function of time, the time average required to (4.12) Td the distribution relaxes and maintains a steady state during open periods. The rate constant reduces to k = k , ( h ) , where k , is the usual steady-state rate constant for an ungated reaction. Both limits are difficult to realize owing to the nature of diffusional relaxation. The spatial distribution rapidly departs from equilibrium during the initial stages of an open cycle, typically in a time of about 10-3~d.This is followed by a much slower relaxation to the steady state in a time on the order of 1037d. Thus K ( T ~ remains ) intermediate between its limiting values k, and k, over a 70 range spanning six orders of magnitude. The approximate expression k N k , ( h ) is also valid in a separate case in which the gate-closed time T , is very small compared to T d . Then the system behaves almost as an ungated system, since the closed cycles are not sufficiently long to allow significant spatial relaxation between open periods. The approximate rate constant k = K(rO) ( h )was further generalized by removing the restriction of fixed gate-open cycle durations so as to allow a distribution of open cycle times. This is necessary to treat real systems where the protein gate opens and closes stochastically. In this paper our primary aim has been to couple the protein gate problem to diffusion for a very simple model of gate dynamics. To apply these results to realistic cases of ligand-protein binding, we must expand our treatment to include several other important features which we now discuss. First, we must develop and incorporate a dynamic description of the gate itself. Progress along this line can be made by identifying an internal coordinate of the protein controlling the binding site accessibility. For instance, the protein can be considered to undergo some collective mode of motion along a gate coordinate 5 on a potential energy surface E(5). The shape of E ( [ )and the nature of the 5 motion (e.g., diffusional, inertial) will determine the lifetime of open and closed gate states. For example, the case considered previously with T, >> 7d >> T~ could occur if E(5) were parabolic with minimum energy E = 0 corresponding to a closed gate and E > kBT for the open state (kBTis Boltzmann’s constant times absolute temperature). The case where TOT(, > Td might result if E ( [ ) had two regions of stability corresponding to open and closed states with an intervening activation barrier. A second aspect of gated reactions we have only briefly mentioned is the possibility of different forms of h(t). For example, the protein reactivity (binding site accessibility) may vary over a continuous range of values as the gate is in the process of opening or closing. If this occurs on the diffusion time scale, a characteristic gate function h ( t ) which varies over an intermediate range of values from 0 to 1 will be required. In order to focus on gating effects on diffusion-influenced reactions, we have omitted the important feature of anisotropic reactivity. In a more complete treatment one must clearly include the effect of dynamically varying orientation of the protein binding site relative to the incoming ligand. The reduction to a single radial coordinate would then no longer be possible. These and other aspects of gated diffusion-influenced reactions, along with possible application to gated membrane c o n d u c t a n ~ e will , ~ ~ be the subject of future work. +
Northrup et al.
the donors of the Petroleum Research Fund, administered by the American Chemical Society (Tennessee Technological University). J.A.M. is an Alfred P. Sloan Fellow and the recipient of an NIH Research Career Development Award. Appendix A Here we sketch the solutions of the Smoluchowski diffusion equations mentioned in section 111. The reactive Green’s function Grx(r,r’,t)for radial diffusion in a region r 2 R in the presence of a spherically symmetric surface sink at r = R with intrinsic reactivity coefficient k , and reflecting wall at r = R is defined as the solution of eq 3.1 in text. It has been shown elsewhere4 that this inhomogeneous (source) equation is equivalent to the homogeneous diffusion equation
supplemented with the radiative boundary condition at surface r = R
and with initial condition Grx(r,rr,O)= 6(r - r’). The solution to eq A.l is given in ref 29 as 1
+
[exp(-(r - rq2/4Dt) 8arr’(aDt)li2 exp(-(r + r’- 2R)2/4Dt) - c(47rDt)’i2exp(c2Dt + (r + r’-2R)c) erfc((r + r’- 2R)/2(Dt)’i2 + c(Dt)’i2)] (A.3)
Grx(r,r’,t)=
where c = R-l(l + y), y = Rk,/D, and erfc is the complementary error function. In the gate-closed state relaxation is governed by the unreactive or reflecting wall Green’s function Grefl(r,r’,t),which satisfies eq A.l but with k, = 0 in the boundary eq A.2. The solution is given by eq A.3 with c = R-’ (i.e., y = 0). The important diffusion problem which must be solved in order to obtain the result of eq 4.7 is p(R,t) resulting from relaxation from an equilibrium initial distribution in a system with reactivity k,. This distribution can be constructed from GJr,rr,t) according to eq 3.2 with p(rl,tl) = 1. Performing this integral we obtain the intermediate result for the full distribution p(r,t) = 1 - y(1 + y)-’(R/r)(erfc [(r - R)/(2(Dt)1/2)]exp[c2Dt + c(r - R)] erfc [(r - R)/(2(Dt)’i2) + ~ ( D t ) l / ~(A.4) ]] It is this distribution evaluated at contact separation r = R that appears in the time-dependent rate coefficient KU(t) for an ungated reaction (with an equilibrium spatial initial condition) and is given in eq 4.2. Appendix B Here we briefly describe the finite difference numerical method used to obtain exact solutions to eq 2.3 for arbitrary h(t). First we define dimensionless variables for spatial separation, z = r/R, and for time, T = Dt/R2. Continuous space and time is then discretized by the definitions z = jSz j = 1, 1 + 1, ..., N T
= iSr
i = 0, 1, ..., m
hz’p(z,~)
+
Acknowledgment. This work has been supported in part by grants from the NSF, the Robert A. Welch Foundation (University of Houston), the Research Corporation, and
(B.1)
Pv,i)
(29) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids”, 2nd ed, Clarendon, Oxford, 1959, p 368.
J. Phys. Chem. 1982, 86, 2321-2324
Here 6z and 8r are small space and time increments (chosen to be 0.05 and 0.0008, respectively). The inner boundary lattice point 1 depends upon the value of 6z chosen. With 6z = 0.05, r = R corresponds to z = 1 and 1 = 20. Equation 2.3 is replaced by P(j,i+l) - Po’,;) = ~ j P ( j , i ) bjP(j+l,i) ~jP(j-l,i); j = 1, ..., N (B.2)
+
+
+
where aj = a [ ( j- 1/2)-l- (j 1/2)-l - 21, bj = a[l - (j + 1/2)-l], cj = a[l (j - 1/2)-l], and a = 6r6z-’. This is supplemented with an inner boundary condition P(1 - 1,i) = P(l,i){-[~i + c ~ - ~ ] / -c LY ~ ( Y ~ z /(B.3) c~)
+
2321
and an equilibrium outer boundary condition P(N,i) = 4 7 ~ ( N 6 Z ) ~The . inner boundary condition contains the time-varying intrinsic reactivity yi (= y when h = 1; = 0 otherwise). The first term in brackets { 1 in eq B.3 enforces the reflecting wall condition when yi = 0. The initial condition in all gated reaction solutions is the equilibrium condition P(j,O) = 4 ~ 0 ’ 6 ~ )A~ more . complete detailed discussion of this method is given e l ~ e w h e r e . ~ ~ ~ ~ ~ (30) S. H. Northrup and J. T. Hynes,J.Chem. Phys., 69,5246 (1978). (31) H. B. Keller in ‘Mathematical Methods for Digital Computers”, A. Ralston and H. S. Wilf, Ed., Vol. 1, Wiley, New York, 1967, p 135. (32) E. Neher and C. F. Stevens, Annu. Reu. Biophys. Bioeng., 6,345 (1977).
Gas-Phase Reactions of 0- and 0,- with a Variety of Halogenated Compounds Gerald E. Streit Los AIamos Scientific Laboratory, Los Alamos, New Mexico 87545 (Received: December 10, 1981; I n Final Form: January 28, 1982)
The flowing afterglow technique has been applied to a comparative study, for a series of halogenated compounds, of dissociative attachment and reaction with 0- and 0;. The ion product spectrum produced by reaction with 0- and 0 2 - overlaps but is more diverse than that obtained by dissociative attachment. Absolute rate constants for the 0- and 02-reactions are reported. A few reactions, for which exothermic mechanisms can be written, were found to be quite inefficient.
Introduction Gas-phase negative-ion chemistry is of interest in such diverse fields as aeronomy,la laser discharges, gas-phase aciditieslb and electron affinities,lcV2negative chemical ionization analysis,ldthermochemistry, and solvation eff e c t ~ . ~The reactions of the 0- and 02-anions are of particular interest in ionospheric ion chemistry and atmospheric-pressure chemical ionization and for making comparisons between gas-phase and solution chemistry. In negative chemical ionization analysis it is very likely that 0- and 02-,if not the reactant ions of choice, either will be intermediates in forming the reactant ion4 or will be contaminant negative ions causing complications in the mass spectrum.ld Halogenated compounds have similarly received substantial recent attention as possible modifiers of atmospheric chemistry and as trace contaminants in the environment. Thus, ion-molecule reactions involving halogenated compounds are of interest in the overall mechanism of atmospheric chemistry and for developing analytic procedures for trace analysis. To date there have been a very limited number of studies of the reactions of 0,- ( x = 1,2,3) with halogenated compounds. Tanaka et al.3 studied the reactions of a number of negative ions including 0- with singly halo-
genated methanes. Their interest was in the comparison of the rates of nucleophilic displacement reactions in the gas phase to the rates in solution. Fehsenfeld et al.5 have examined the reactions of 02-with CFCl, and with CFzC12 in order to assess the magnitude of a tropospheric or stratospheric sink for those halocarbons. Dotan et al.6have examined the reactions of 0- with CC14and of 0- and Ozwith HC1 with regard to the role of chlorinated compounds in atmospheric negative-ion chemistry. Siegel and Fite’ looked at the negative ions produced by atmosphericpressure ionization of several of the common fluorochlorocarbons and saw C1- as the only negative ion. However, the carrier gas used was not specified, so they may have observed only the dissociative attachment process rather than a charge-transfer or reactive process. An interesting application of Oz- chemistry has been made by Grimsrud and Millersa in demonstrating that oxygen doping of the carrier gas in gas chromatography significantly enhances the electron capture (EC) detector sensitivity to monohalogenated methanes. This has been extended by Phillips et al.,sbwho utilize the chemistry of 0- following dissociative attachment to NzO to enhance the electron capture detector response to compounds which do not directly attach electrons. In this study the reactions of 0- and 0,- with a large number of halogenated compounds have been examined
(1) (a) E. E. Ferguson, F. C. Fehsenfeld, and D. L. Albritton in ‘Gas Phase Ion Chemistry”, M. T. Bowers, Ed., Academic Press, New York, 1979, Vol. 1, p 45; (b) J. E. Bartmess and R. T. McIver, Jr., ibid., Vol. 2, p 88; (c) B. K. Janousek and J. I. Brauman, ibid.,Vol. 2, p 53; (d) K. R. Jennings, ibid., Vol. 2, p 124. (2) G . E. Streit, J. Chem. Phys., in press. (3) K. Tanaka, G. I. Mackay, J. D. Payzant, and D. K. Bohme, Can. J. Chem., 64, 1643 (1976). (4) A. L. C. Smit and F. H. Field, J . Am. Chem. SOC.,99, 6471 (1977).
(5) F. C. Fehsenfeld, P. J. Crutzen, A. L. Schmeltekopf, C. J. Howard, D. L. Albritton, E. E. Ferguson, J. A. Davidson, and H. I. Schiff, J . Geophys. Res., 81, 4454 (1976). (6) I. Dotan, D. L. Albritton, F. C. Fehsenfeld, G. E. Streit, and E. E. Ferguson, J . Chem. Phys., 68, 5414 (1978). (7) M. W. Siegel and W. L. Fite, J . Phys. Chem., 80, 2871 (1976). (8) (a) E. P. Grimsrud and D. A. Miller, Anal. Chem., 50, 1141 (1978); (b) M. P. Phillips, R. E. Sievers, P. D. Goldan, W. C. Kuster, and F. C. Fehsenfeld, ibid., 51, 1819 (1979).
0022-3654/82/2086-2321$01.25/00 1982 American Chemical Society
