It is shown that the rates of diffusion-controlled processes may have a solvent vicosity independent part as well as a viscosity dependent part. Some relevant experiments involving intramolecular polypeptide movements are discussed, and implications for some experiments on diffusion in membranes are outlined

Weaver, D. L.

English

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NASA CONTRACTOR REPORT 166394 
Viscosity Dependence of the 
Rates of Diffusional Processes 
David L. Weaver 
NASA Cooperative Agreement NCA 2-0R785-210 
August 1982 
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NASA CONTRACTOR REPORT 166394 
Viscosity Dependence of the 
Rates of Diffusional Processes 
David L. Weaver 
Department of Physics 
Tufts University 
Medford, Massachusetts 
Prepared for Ames Research Center under 
NASA Cooperative Agreement NCA 2-0R785-201 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 

1. 
INTRODUCTION 
Systems which are overdamped due to the frictional drag of solvent 
have the motion of their" constituents described by the diffusion equation 
as an approximation to the Fokker-P1anck equation (1). Because of the 
frequent occurrence of this condition, diffusional models are commonly 
used to describe biological, physical and chemical processes including 
heat conduction, mass transport, some aspects of chemical reactions, 
and so on. In order to apply the general diffusion equation to a particular 
physical situation or model, the space in which diffusion occurs must be 
designated and the behavior of appropriate combinations of the particle 
density and its derivatives specified at the boundaries. However, in the 
physical situation in which there is partial or complete absorption 
(capture or reaction depending on the particular problem being modeled) 
at the boundary, it has not been completely clear how to interpret the 
parameters determining the boundary condition. In Ref. 2 in the context 
of the diffusion-collision model of protein folding, two possible 
parameter choices were discussed in detail and a third possibility, 
the one to be outlined below in the present paper, was mentioned without 
proof. Since the third possibility leads to the rates of diffusion-
controlled processes having a solvent viscosity-independent part as 
well as a viscosity-dependent part, it has important consequences 
for intramolecular and intermolecular movements in biological systems. 
In addition, recent numerical studies (3) and approximate analytical 
calculations (4) on one-dimensional diffusional systems have lent 
credence to the third possibility. Therefore, the purpose of the 
present paper is to present a derivation of the "radiation" or partially 
absorbing boundary condition (the perfect absorber being a special 
case) for a diffusion-controlled process in which not every collision 
2. 
between a Brownian particle and the absorb~ng boundary results in 
removal of the particle from the diffusion space. The details of 
the derivation are follwed by a discussion of some experiments involving 
intramolecular polypeptide movements in which the solvent viscosity 
was varied, and the implications for diffusion in membranes in which 
localization due to trapping takes place (5,6). 
THE DERIVATION 
To be specific, one-dimensional diffusion will be discussed below. 
The designation of the space determines whether the system is infinite 
or finite, and in the latter case one must specify the length of 
. the space in order to calculate quantities of physical interest in 
absolute terms. 
Consider finite space diffusion first (infinite space diffusion 
will be considered later). Boundary conditions must be specified 
at two boundaries, x = a and x = b with b > a. Let the particle density 
be p(x,t) at position x and time t. The diffusion equation is 
~=Da2g 
at ax a < x < b 
(1) 
with D the diffusion coefficient (assumed constant). The boundary 
conditions at a and b, which along with the initial value p(x,o) are 
needed to specify p(x,t), are to be linear combinations of p and its 
first derivatives ap/ax and ap/at. In the absence of particle emission 
at the boundaries, ap/at will not contribute to the boundary conditions. 
A 
Since the particle flux is a(x,t) = -Dap/ax ~, which is the net number 
of particles per second crossing a unit area perpendicular to the x-axis, 
a completely reflecting boundary at which no absorption takes place is 
uniquely specified by the condition a = O. Let this boundary condition 
prevail at x = b in the one-dimensional finite space example. Thus, 
op/ox = 0 at x = b. 
At the boundary x = a, let the conditions be such that partial 
absorption takes place. That is, some fraction of the Brownian particles 
that reach the boundary at x = a are not reflected and no longer diffuse. 
This general case encompasses the limits of perfect absorption and 
perfect reflection as well as all intermediate situations. For partial 
absorption, the reflected flux at x = a will be smaller than the incident 
flux by the number of particles per second being absorbed, that is 
0+ • yo_, 0 ~ y ~ 1, where 0_ is the magnitude of the flux toward 
the boundary at x = a and d+ is the magnitude of the reflected flux 
at x = a (that is, the portion of the net flux at x = a going in the 
positive x direction). In the spirit of kinetic theory, one may write 
.o%(a,t) = <v>%P%(a,t) where P±(a,t) are the reflected (+) and incident 
(-) particle densities and <v>± are "average" values for the reflected 
and incident particle velocities at x = a derived from the mean value 
theorem of the calculus. If the absorption mechanism at the boundary 
is independent of velocity, then <v>+ = <v> = <v>. This will be 
assumed as the simplest possibility. In many situations the boundary 
will be due to a potential energy barrier whose height sets a minimum 
value for the kinetic energy of the incident particles in order to 
be absorbed at x = a. Nevertheless, if the approach to a Maxwell-
Boltzmann velocity distribution is fast on the diffusion time scale, 
3. 
then the incident and reflected particle velocity distributions will be 
virtually identical. Then, since 
and 
one obtains the boundary condition 
or 
a(a,t) = -(1 - y) (1 + y) <v> p(a,t) 
~ I = (1 - y) ~ I 
ax- (1 + y\ D P 
a ' a 
Note that with this boundary condition, perfect absorption y = 0 does not 
imply pi = O. However, for perfect reflection y = 1, the condition 
a 
ap/axla = 0 is found. The consequence of Eq. (5) for several biophysical 
situations have been briefly mentioned by the present author previously 
(2,6,7) without discussion. Below some of the implications and relevant 
experiments are analyzed. 
THE IMPLICATIONS 
4. 
(2) 
(3) 
(4) 
(5) 
Eq. (5) has important implications about the rates of some diffusional 
processes. The rates of absorption in the model outlined above may be 
calculated using an extension of the idea of first passage times (8), as 
shown by a number of authors (9,10,11). This is carried out below for 
the diffusional model considered above by writing the fraction of particles 
absorbed at x = a in terms of the mean absorption time (inverse of the 
absorption rate. 
Let the fraction of absorbed 
b 
n.a ( t) - 1 - f a dx p( x, t ) 
particles be n (t), so that 
a 
5. 
where p is normalized initially to one particle (thus, p may be thought 
of as the relative position probability density). Solution of the 
diffusion equation with the given boundary conditions and an initial 
distribution in which the particles start at x = b indicates that the 
infinite series of decreasing exponential terms which represents the 
(6) 
time dependence of n (t) may be well-approximated by a single exponential. 
a 
That is, 
n (t) 
a 
III 
= I - exp( -t!-d 
with 
T -
(b-a)2 + (b-a) 
2D <v> 
(I+y) 
(II) 
If the initial particle distribution is uniform rather than concentrated 
at x = b, Eq. (8) is modified slightly be the replacement ?f 2 by 3 in 
the first term. 
Two points are indicated by Eq. (8)~ First, that a measurement of 
the inverse rate should show a linear dependence on solvent viscosity 
(via D) plus a viscosity independent term (the other forms of Eq. (5) 
which are commonly used (2) would have both ~rms in Eq. (8) being 
-1 proportional to D ). Second, that a perfect absorber (y = 0) has a 
viscosity independent term, which can complicate the use of diffusion 
measurements to find D (see below). As mentioned in Refs. 3 and 4, the 
y - 0 perfect absorber situation does not (using Eq. (5» lead to 
(7) 
(8) 
pi - 0 as the boundary condition. It does, however, lead to the boundary 
a 
condition p(x,t) '"' 0 at x '"" a - 6, 6 > 0 where 6 is a length parameter 
(referred to as the Milne extrapolation length) whose value is 
6. 
fl _ (1+X) D 
(1-X) <v> (9) 
Eq. (9~ was found by computing the rate of particle injection at x = b 
needed to have the above diffusional model at steady-state with one particle 
in the region (a,b) in order to compare with the results of Refs. 3 and 4. 
This comparison also allows one to more precisely identify <v> to be 
(a/2)! where a ~ kBT/m with kB the Boltzmann constant, T the absolute 
temperature and m the Brownian particle mass. Note that a is the mean 
square velocity in one-dimension for a Maxwell-Boltzmann distribution. 
Another interpretation is <v> = vM/2 where vM is the most probable 
speed. 
SOME RELEVANT EXPERIMENTS 
The "radiation" boundary condition derived above leads to the possibility 
that the rates of diffusion driven processes may not be directly pro-
portional to D and, in fact, may be effectively independent of D for 
small enough x. In this regard, some experiments have been carried out 
to study the intramolecular motion of polypeptide chain segments in 
solvents of different viscosities (since, according to the Einstein 
relation D = kBT/f with f the friction coefficient, varying the viscosity 
ought to vary D in an inverse manner). Haas et a1. (12) investigated the 
kinetics of the fluorescence decay of the energy donor in a homologous 
series of oligopeptides, each containing at its ends a donor and an 
acceptor of electronic excitation decay energy in solvent mixtures of 
different viscosities. With.an assumed theoretical analysis, diffusion 
coefficients were derived which increased systematically upon decreasing 
7. 
the solvent viscosity. The values obtained for the diffusion coefficients 
were about an order-of-magnitude smaller than the values expected for the 
diffusion coefficients of the free chromophores in solvents of comparable 
viscosity, and appear to have a solvent viscosity independent part in 
the derived values of D-I • Although the results obtained for D in this 
experiment are probably model dependent, there is a clear dependence 
of diffusion coefficient on solvent viscosity, as well as the suggestion 
-1 
of a viscosity independent contribution to D • In the analysis of this 
experiment the boundary condition p (a,t) = 0 was used rather than 
Eq. (5) with y = o. 
Tsong and Baldwin (13), on the other hand, in their study of the 
kinetics of folding of the two forms of unfolded ribonuclease A (with all 
disulfide bonds intact) as a function of solvent viscosity, by adding 
either sucrose or glycerol, found no dependence on solvent viscosity, 
the rates of the two observed reactions being either unchanged or slightly 
faster in the presence of sucrose or glycerol. In the same system, 
Tsong (14) has recently found a reaction which is strongly dependent 
on solvent viscosity and somewaht faster than the reactions observed 
in Ref. 13. 
The precise interpretation of these experimental results is not 
completely clear, at present, due to the complexity of the systems 
involved. However, it appears that diffusion mediated reactions are 
playing a signfiicant role, and that there are both solvent viscosity 
dependent and solvent viscosity independent contributions to the kinetics, 
perhaps of the kind suggested by Eq. (8) and the subsequent discussion. 
If one assumes that considerations similar to those leading to 
Eq. (5) apply also to the two-dimensional surface of membranes, then 
8. 
localization in such systems due to diffusion-driven trapping (5,6) 
will a!so show the mixed dependence on D of Eq. (B) ,although the detailed 
form will be different (see Ref. 6). In that case, D would not be 
simply directly proportional to the localization rate. This problem 
requires further study. Some approximate results have been obtained 
by Harris (15) for the related problem of steady absorption of Brownian 
particles by an infinite right circular cylinder, corresponding to 
absorption by a circular sink in two dimensions. A modified boundary 
condition was again found by introducing an effective radius of absorption 
corresponding to the Milne extrapolation length in one-dimension. 
The extension of the above ideas to infinite systems leads to 
further interesting results concerning approach to equilibrium effects. 
Consider, as a definite example, a spherically symmetrical infinite 
system of molecules of initial uniform concentration p surrounding 
o 
a target of radius a centered at r = o. The molecules diffuse in 
the infinite space, and their concentration as a function of position 
and time, p(r,t) satisfies the diffusion equation, Eq. (1), within 
the limits a < r < m. Corresponding to Eq. (5), the boundary condition 
appropriate to the infinite space is 
aL ~ I = pi ar a 
a 
where aL will be the three-dimensional analog of (1 + Y)D/«l - Y)<v» 
in Eq. '(5). As r ~ m, p(r,t) ~ p for the other boundary condition. 
o 
The above diffusion problem may be solved, in principle, using 
. 2 a ' 
the Laplace transform method to obtain .4na D ~ la' the time dependent 
rate of association. To proceed further, suppose that the initial 
(10) 
concentration of targets is equal to p also. Then, the theory of 
o 
von Smoluchowski (16) for colloidal particles may be applied here 
to yield n (t) the probability that a target molecule pair is formed 
a 
before time t. The result is 
(t/[ (1 + L)TJ) 
°a(t) - 1 + (t/[(l+L)T]) 
where T E 1/(41TaDp ), the "unit" of time appropriate to this physical 
o 
situation. Again, there appears a mixed D dependence in the overall 
rate of association. Harri~ (17) has recently studied numerically 
the related problem of steady-state absorption by a sphere. 
9. 
(11) 
10. 
REFERENCES 
1. Chandrasekhar,S.,1943. Stochastic problems in physics and astronomy. 
Rev.Mod.Phys. 15: 1-89 • 
. 
2. Karp1us,M. and D.L.Weaver,1979. Diffusion-collision model for protein 
folding. Biopo1ymers 18: 1421-1437. 
3. Burschka,M.A. and U.M.Titulaer,198l. The kinetic boundary layer for 
the Fokker-Planck equation with absorbing boundary. J.Stat.Phys. 1i: 
569-582. 
4. Harris,S.1981. Steady,one-dimensiona1 Brownian motion with an absorbing 
boundary. J.Chem.Phys. 1l: 3103-3106. 
5. Chao,N.M., S.H. Young and M.M.Poo,198l. Localization of cell membrane 
components by surface diffusion into a trap. Biophys.J.~: 139-153 
6. Weaver,D.L.,1982. Diffusion mediated localization on membrane surfaces. 
Biophys.J. (in press). 
7. Weaver,D.L.,1980. Diffusion and the "fast-exchange" model. J.Mag.Res. 
37: 543-546. 
8 Weiss,G.H.,1967. First passage time problems in chemical physics. 
Adv.Chem.Phys. 13: 1-17. 
9. Weaver,D.L.,1979. Some exact results for one-dimensional diffusion with 
"absorption. Phys.Rev. B. 20: 2558-2561. 
10. Szabo,A., K.Schulten and Z.Schulten,1980. First passage time approach 
to diffusion controlled reactions. J.Chem.Phys.ll: 4350-4357. 
11. Deutch,J.M.,1980. A simple method for determining the mean passage 
time for diffusion controlled processes. J.Chem.Phys. 11: 4700-4701. 
12. Haas,E., E.Katcha1ski-Katzir and I.Z.Steinberg,1978. Brownian motion 
of the ends of oligopeptide chains in solution as estimated by energy 
transfer,between the chain ends. B1opo1ymers 17: 11-31. 
13. Tsong,T.Y. and R.L.Ba1dwin,1978. Effects of solvent viscosity and 
different guanidine salts on the kinetics of ribonuclease A chain folding. 
Biopo1ymers 17: 1669-1678. 
14. Tsong,T.Y. (private communication). 
15. Harris,S.,1982. Steady absorption of Brownian particles by an infinite 
cylinder. J.Chem.Phys.~: 587-592. 
16. Von Smoluchowski,M.,19l7. 
Zeit.F.physik Chemie 92: 129 
17. Harris,S.,1982. Steady absorption of Brownian particles by a sphere. 
J.Chem.Phys. ]2: 934-938. 
1. Report No. 2. G~ment Acc:asion No. 3. Recipient" Catalog No. 
NASA CR-166394 
4. Tltll 8I1d Subtitll 5. Report Date 
Viscosity Dependence of the Rates of August 1982 
Diffusional Processes 
8. Performing Organization Code 
7. Authorls) S. Performing Organization Report No. 
David L. Weaver 
10. Work Unit No. 
9. P8rforming Organization Name and Address T4568 
Department of Physics 11. Contract or Grant No. 
Tufts University NCA 2-0R785-201 
Medford MA 02155 13. Type of Report and Period Covered 
12. Sponsoring Agency Name .,d Address Contractors Report 
National Aeronautics and Space Administration 14. Sponsoring Agency Code 
Washington, D.C. 24056 199-60-62 
15. Supplementary Notes 
Robert D. Mac Elroy , Technical Monitor, Mail Stop 239-4, Ames Research Center, 
Moffett Field, CA 94035 (415) 965-5573 FTS 448-5573. 
16. Abstract 
It is shown that the rates of diffusion-controlled processes may have a 
solvent viscosity independent part as well as a viscosity dependent part. 
Some relevant experiments invo1v~ng intramolecular polypeptide movements 
are discussed, and implications for some experiments on diffusion in 
membranes are outlined. 
17. Key Words ISuggested by Authorlsll 18. Distribution Statement 
Diffusion Control 
Viscosity Dependence Unclassified - Unlimited 
Protein Folding STAR Catergory 51 
19. Security Oaaif. lof this report) 20. Security Claw!. lof thi' paga) 21. No. of Pages 22. Price· 
Unc1ass if ied Unclassif ied 13 
-For sail by the National Technical Information Service. Springfield. Vil"\lini. 22161 
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Viscosity dependence of the rates of diffusional processes

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