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Diffusion and the “fast-exchange” model
JOURNAL
OF MAGNETIC
37, 543-546 (1980)
RESONANCE
COMMUNICATIONS
Diffusion and the “Fast-Exchange” Model* A model widely used in the interpretation of proton magnetic resonance in water in biological samples is the so-called “fast-exchange” model in which water diffuses freely until it relaxes in some way at a boundary or within the system volume. In the “fast-exchange” model, a single relaxation time dominates the relaxation kinetics, and Brownstein and Tarr (1) have carried out a derivation of this model using the classical diffusion equation with appropriate boundary conditions and volume relaxation rate. The purpose of this communication is to clarify the interpretation of the “fast-exchange” model via diffusional kinetics, particularly as discussed in Ref. (1). The first point to make is that when one takes the limit as the diffusion coefficient D+oo, one is removed from the regime in which the diffusion equation in position space is valid. It is known (See, e.g., (2)) that the diffusion equation in position space corresponds to a large friction coefficient (small diffusion coefficient) so that the motion is overdamped with respect to the friction drag. The second point is that the single relaxation time approximation is well defined in terms of the mean relaxation time 7 defined below. Let p(x, x0, t) be the probability density of finding a magnetic moment at position x, at time f, if it had been at x0 at t = 0. Then, the total magnetic moment in the diffusional volume (“free” phase) is N(xo, t) = The mean relaxation time r is then defined by -r/r N(xo, t) = e
El
with N(xo, 0)= 1 for convenience. The single exponential approximation to N(xo, 1) has the same initial and final values in time as the true N(xo, t) and the same area under the N versus t curve when T itself is given by cc
dt l-j [I -N(xo, t)]
dxo)= cc = I
0
dt iV(xo, t).
By this definition, 7(x0) is an extension of the concept of the “first-passage” time (3) to situations in which relaxation at a boundary of the system does not occur with * Supported in part by the National Science Foundation under Grant PCM-7807633. 0022-2364/80/03OS43-04%02.00/O 543
Copyright @ lVg0 by Academic Press. Inc. All rights of reproduction in any form reserved. Printed in Great Britain
544
COMMUNICATIONS
every collision and/or in which relaxation may occur within the diffusional volume itself. Such an extension has been outlined by Weaver (4) and by Szabo et al. (5). As shown in Ref. (4), when p(x, ~0, t) satisfies the diffusion equation in x, t, N(xo, t) satisfies the adjoint partial differential equation in x0, t and therefore 7(x0) obeys an inhomogeneous differential equation in x0 which assuming a constant diffusion coefficient D has the form V&y-=-I/D
[41
with y the volume relaxation rate. Equation [4] may be solved for r in many cases in which complete or partial solution of the diffusion equation requires numerical analysis. For example, consider three cases in which there is radial symmetry, space between r = a and r = b, a two-dimensional space between two concentric cylindrical shells with radii a and b, respectively, and a three-dimensional space between two concentric spherical shells with radii a and b. Then, with a uniform initial distribution, y = 0 everywhere and boundary conditions
ap ar-- 0,
r=b
ap -=ar
r=a
b 16 ”
[51
so that there is a partial relaxation at r = a (here 0 5 p 5 1,0 5 6 I 1 with p and 6 the relaxation and reflection coefficients, and 1 a length parameter whose size depends on the specific mechanism of relaxation at the boundary as discussed below) one obtains the following results: One-dimensional:
T=T(b-u)+-DP
(b-a)* 30
Two-dimensional:
16 (6*-a*) 7=Dp 2u
b4 In b/a +2D(b2-a*)+
Three-dimensional:
IS (b3-u3) “Op 3a2
bh +3Du(b3-u3)
[61
’ (a*-3b2)) 8D
’
[71
[81
The first term in each of the three cases in Eqs. [6], [7], and [8] has the general form --IS v Dfi A’
[91
where V is the “volume” of the diffusion space and A is the “surface area” of the target (with appropriate interpretation for the different dimensions). This is the form found in Ref. (1) by an approximation. The remaining terms in Eqs. [6], [7], and [8] represent the mean relaxation time when the kinetics are strictly diffusioncontrolled, that is, when each encounter with the boundary at r = a results in
54.5
COMMUNICATIONS
relaxation (p + 1, S + 0). So, the situations in which Eq. [9] represents the relaxation time occur when the rate of relaxation is dominated by a small probability of relaxation during a collision at a boundary of the system rather than by the time for the diffusing species to transverse the diffusion space once. It is interesting to note that a similar model has proved valuable in protein folding dynamics (6). When the volume relaxation rate y is not zero, the mean relaxation time contains additional contributions. For example, suppose that y = constant within the diffusion space and consider the one-dimensional model discussed above. Then, in place of Eq. [6], one obtains PDll~ i-Z---1 y ~‘(6 - a)[1 + (p/IS)(D/y)“’ coth (y/D)“*@
-a)]’
i.101
To check this result, one notes that as /3 +O, 6 + 1 so that there is no relaxation at the boundary, r + l/ y as expected. Also, as y + 0, the mean relaxation time reduces to the result quoted in Eq. [6]. Equation [lo] shows that in general, the volume and boundary relaxation rates are coupled in a nonlinear way and do not simply add to produce 7, as found in Ref. (1) for the smallest eigenvalue of the diffusion equation. The length parameter I which occurs in the boundary condition at a partially relaxing boundary has a variety of interpretations depending on the relaxation mechanism. For example, if there is a high potential energy barrier at r = a, then 1 is proportional to the width of the barrier (6). If the boundary has regions in which relaxation occurs surrounded by regions in which no relaxation occurs, then 1 is proportional to the size of the relaxing regions (7), assumed to be small. A third possibility is that 1 -D/v, where v is the root-mean-square speed at temperature T, that is, 1 is the mean free path. In this case, T as approximated by Eq. [9] would be independent of the diffusion coefficient. In summary, when a single relaxation time dominates the kinetics of relaxation in water in biological samples, the mean relaxation time describes the kinetic behavior. This behavior does not imply D+co, however, and in some cases the mean relaxation time is independent of D whereas in other cases it is inversely proportional to D, the particular dependence being related to the mechanism of relaxation. The “fastexchange” model validity, therefore, depends on factors other than the size of D such as a boundary with a small probability of relaxation (in the y = 0 case). When there is no absorption at a boundary and y f 0 but constant, r = l/y, again independent of D. Finally, in the boundary-controlled “fast-exchange” model when Eq. [9] describes the relaxation time adequately, the term “fast” is a misnomer, since the relaxation time is considerably larger than the time to diffuse once through the volume in question. REFERENCES 1. K.R.BROWNSTEINANDC.E.TARR,J. Magn. Reson. 26, 17 (1977). 2. S. CHANDRASEKHAR, Rev.Mod. Phys. 15, l(l943). 3. G. H. WEISS, Adu. Chem. Phys. 13, 1 (1967). 4. D. L. WEAVER, Phys. Rev. B, 20, 2558 (1979).
546
COMMUNICATIONS
5. A. SZABO, K. SCHULTEN,AND~.~HULTEN, preprint. 6. M.KARPLUSANDD.L. W~~~~~,BiopoIymers l&1421(1979). 7. H.C.BERGANDE.M.PURCELL,E~O~~~S.J. 20,193 (1977).
D.L. Department of Physics Tufts University Medford, Massachusetts 02155 Received August 10, 1979
WEAVER
OF MAGNETIC
37, 543-546 (1980)
RESONANCE
COMMUNICATIONS
Diffusion and the “Fast-Exchange” Model* A model widely used in the interpretation of proton magnetic resonance in water in biological samples is the so-called “fast-exchange” model in which water diffuses freely until it relaxes in some way at a boundary or within the system volume. In the “fast-exchange” model, a single relaxation time dominates the relaxation kinetics, and Brownstein and Tarr (1) have carried out a derivation of this model using the classical diffusion equation with appropriate boundary conditions and volume relaxation rate. The purpose of this communication is to clarify the interpretation of the “fast-exchange” model via diffusional kinetics, particularly as discussed in Ref. (1). The first point to make is that when one takes the limit as the diffusion coefficient D+oo, one is removed from the regime in which the diffusion equation in position space is valid. It is known (See, e.g., (2)) that the diffusion equation in position space corresponds to a large friction coefficient (small diffusion coefficient) so that the motion is overdamped with respect to the friction drag. The second point is that the single relaxation time approximation is well defined in terms of the mean relaxation time 7 defined below. Let p(x, x0, t) be the probability density of finding a magnetic moment at position x, at time f, if it had been at x0 at t = 0. Then, the total magnetic moment in the diffusional volume (“free” phase) is N(xo, t) = The mean relaxation time r is then defined by -r/r N(xo, t) = e
El
with N(xo, 0)= 1 for convenience. The single exponential approximation to N(xo, 1) has the same initial and final values in time as the true N(xo, t) and the same area under the N versus t curve when T itself is given by cc
dt l-j [I -N(xo, t)]
dxo)= cc = I
0
dt iV(xo, t).
By this definition, 7(x0) is an extension of the concept of the “first-passage” time (3) to situations in which relaxation at a boundary of the system does not occur with * Supported in part by the National Science Foundation under Grant PCM-7807633. 0022-2364/80/03OS43-04%02.00/O 543
Copyright @ lVg0 by Academic Press. Inc. All rights of reproduction in any form reserved. Printed in Great Britain
544
COMMUNICATIONS
every collision and/or in which relaxation may occur within the diffusional volume itself. Such an extension has been outlined by Weaver (4) and by Szabo et al. (5). As shown in Ref. (4), when p(x, ~0, t) satisfies the diffusion equation in x, t, N(xo, t) satisfies the adjoint partial differential equation in x0, t and therefore 7(x0) obeys an inhomogeneous differential equation in x0 which assuming a constant diffusion coefficient D has the form V&y-=-I/D
[41
with y the volume relaxation rate. Equation [4] may be solved for r in many cases in which complete or partial solution of the diffusion equation requires numerical analysis. For example, consider three cases in which there is radial symmetry, space between r = a and r = b, a two-dimensional space between two concentric cylindrical shells with radii a and b, respectively, and a three-dimensional space between two concentric spherical shells with radii a and b. Then, with a uniform initial distribution, y = 0 everywhere and boundary conditions
ap ar-- 0,
r=b
ap -=ar
r=a
b 16 ”
[51
so that there is a partial relaxation at r = a (here 0 5 p 5 1,0 5 6 I 1 with p and 6 the relaxation and reflection coefficients, and 1 a length parameter whose size depends on the specific mechanism of relaxation at the boundary as discussed below) one obtains the following results: One-dimensional:
T=T(b-u)+-DP
(b-a)* 30
Two-dimensional:
16 (6*-a*) 7=Dp 2u
b4 In b/a +2D(b2-a*)+
Three-dimensional:
IS (b3-u3) “Op 3a2
bh +3Du(b3-u3)
[61
’ (a*-3b2)) 8D
’
[71
[81
The first term in each of the three cases in Eqs. [6], [7], and [8] has the general form --IS v Dfi A’
[91
where V is the “volume” of the diffusion space and A is the “surface area” of the target (with appropriate interpretation for the different dimensions). This is the form found in Ref. (1) by an approximation. The remaining terms in Eqs. [6], [7], and [8] represent the mean relaxation time when the kinetics are strictly diffusioncontrolled, that is, when each encounter with the boundary at r = a results in
54.5
COMMUNICATIONS
relaxation (p + 1, S + 0). So, the situations in which Eq. [9] represents the relaxation time occur when the rate of relaxation is dominated by a small probability of relaxation during a collision at a boundary of the system rather than by the time for the diffusing species to transverse the diffusion space once. It is interesting to note that a similar model has proved valuable in protein folding dynamics (6). When the volume relaxation rate y is not zero, the mean relaxation time contains additional contributions. For example, suppose that y = constant within the diffusion space and consider the one-dimensional model discussed above. Then, in place of Eq. [6], one obtains PDll~ i-Z---1 y ~‘(6 - a)[1 + (p/IS)(D/y)“’ coth (y/D)“*@
-a)]’
i.101
To check this result, one notes that as /3 +O, 6 + 1 so that there is no relaxation at the boundary, r + l/ y as expected. Also, as y + 0, the mean relaxation time reduces to the result quoted in Eq. [6]. Equation [lo] shows that in general, the volume and boundary relaxation rates are coupled in a nonlinear way and do not simply add to produce 7, as found in Ref. (1) for the smallest eigenvalue of the diffusion equation. The length parameter I which occurs in the boundary condition at a partially relaxing boundary has a variety of interpretations depending on the relaxation mechanism. For example, if there is a high potential energy barrier at r = a, then 1 is proportional to the width of the barrier (6). If the boundary has regions in which relaxation occurs surrounded by regions in which no relaxation occurs, then 1 is proportional to the size of the relaxing regions (7), assumed to be small. A third possibility is that 1 -D/v, where v is the root-mean-square speed at temperature T, that is, 1 is the mean free path. In this case, T as approximated by Eq. [9] would be independent of the diffusion coefficient. In summary, when a single relaxation time dominates the kinetics of relaxation in water in biological samples, the mean relaxation time describes the kinetic behavior. This behavior does not imply D+co, however, and in some cases the mean relaxation time is independent of D whereas in other cases it is inversely proportional to D, the particular dependence being related to the mechanism of relaxation. The “fastexchange” model validity, therefore, depends on factors other than the size of D such as a boundary with a small probability of relaxation (in the y = 0 case). When there is no absorption at a boundary and y f 0 but constant, r = l/y, again independent of D. Finally, in the boundary-controlled “fast-exchange” model when Eq. [9] describes the relaxation time adequately, the term “fast” is a misnomer, since the relaxation time is considerably larger than the time to diffuse once through the volume in question. REFERENCES 1. K.R.BROWNSTEINANDC.E.TARR,J. Magn. Reson. 26, 17 (1977). 2. S. CHANDRASEKHAR, Rev.Mod. Phys. 15, l(l943). 3. G. H. WEISS, Adu. Chem. Phys. 13, 1 (1967). 4. D. L. WEAVER, Phys. Rev. B, 20, 2558 (1979).
546
COMMUNICATIONS
5. A. SZABO, K. SCHULTEN,AND~.~HULTEN, preprint. 6. M.KARPLUSANDD.L. W~~~~~,BiopoIymers l&1421(1979). 7. H.C.BERGANDE.M.PURCELL,E~O~~~S.J. 20,193 (1977).
D.L. Department of Physics Tufts University Medford, Massachusetts 02155 Received August 10, 1979
WEAVER