Nonlinear diffusion in biological systems

Bulletin of Mathematical Biology, Vol. 41, pp. 151-162 Pergamon Press Ltd. 1979. Printed in Great Britain © Society for Mathematical Biology

0007M985/79/0301~0151 $02.00/0

NONLINEAR DIFFUSION IN BIOLOGICAL SYSTEMS

• S. H. LIN Department of Chemical Engineering, Polytechnic Institute of New York, Brooklyn, New York 11201, U.S.A.

Diffusion problem with variable diffusion coefficient in a spherical biological system is investigated. Also included in this study is the biological reaction of the Michaelis-Menten type. The problem formulated consists of a highly nonlinear differential equation which, however, can be efficiently solved by the orthogonal collocation method on a digital computer. The effects of the dimensionless governing parameters on the transient and steady state concentration responses are parametrically examined for the diffusion system with and without biological reaction.

1. Introduction. Diffusion with or without reaction occurs in a wide variety of biological, engineering and physical sciences. In the past, the biological diffusion-reaction systems have been extensively investigated by Auchmuty and Nicolis (1975), Hearon (1965, 1968, 1973), Herschkowitz- Kaufman (1975), Lin (1976), Martinez (1972), Massaro and Fatt (1969), Ngian and Lin (1976), Rashevsky (1960), Rosen (1974, 1975), Segel and Jackson (1972), Stetson et al. (1977) and Zhabotinsky and Zaikin (1973). The diffusion problems with or without ordinary chemical reaction have also been extensively examined and solutions to a large class of problems are available (Crank, 1975). However, the previous investigations were mainly concerned with diffusion with constant diffusion coefficient. In a number of real situations, the diffusion coefficient of solute shows dependence on its concentration and hence on the spatial coordinate. Such a variation in diffusion coefficient will render the solution of the diffusion problem more complicated because of appearance of nonlinear terms in the governing material balance equation. This problem has been receiving increasing amount of attention in recent years. Hays and Curd (1967), Mingle (1973), Crank (1975) and Cable and Cardew (1977) have suggested

151

152 S . H . LIN

different approximate or numerical methods for solving the nonlinear diffusion problems without reaction. However, because of the complexity of the problems, no single method can satisfactorily handle the problems under different situations.

The biological counterpart of this diffusion problem with variable diffusion coefficient also exists. Experimental evidences reported by Charm (1973), Dick (1959) and Vaccarezza et al. (1974) have indicated that the diffusion coefficients of water and oxygen are not constant in a number of biological media. The variable diffusion coefficient of oxygen is of parti- cular interest because oxygen is an important substrate involved in many biological oxidation reactions. In spite of the existence of this type of biological diffusion problem, no literature specifically addressing to this problem has been available. The present work represents an attempt to deal with this diffusion problem. For generality, the biological reaction of the Michaelis-Menten type is also included. The results of the present study may provide some relevant information regarding the effect of variable diffusion coefficient on the solute distribution under this type of nonlinear kinetics.

2. Physical Model. The transient, one-dimensional diffusion in a large spherical body with variable diffusion coefficient and the Michaelis-Menten reaction can be described by the following equation:

c3C_1 D [D(C)r20C ] VC . (1) c3t r 2 Or L d~-r C + k~'

and the initial and boundary conditions are given by

t=0 ; C = 0 (2)

r = 0 ; ~ r =0 (3)

r=R; D(C)~r=h(Co-C ) (4)

where C is the concentration, Co the concentration in the bulk phase, D(C) the concentration-dependent diffusion coefficient, V the maximum reaction rate, k,, the Michaelis constant, h the membrane permeability, R the radius, r the radial coordinate and t the time. If the diffusion coefficient, D(C), is a constant, (1) to (4) reduce to those considered by Lin (1976).

NONLINEAR DIFFUSION IN BIOLOGICAL SYSTEMS 153

Different types of concentration-dependen t functions of diffusion coefficient have been suggested (Crank, 1975). For the present work, the following two types are considered"

and

D(C) = Doll +6(C/Co)]

D(C) = D O exp(6 C/Co),

(5)

(6)

in which Do represents a constant diffusion coefficient at very low concentration and 6 a constant parameter. Equations (5) and (6) represent typical linear and nonlinear concentration dependences of diffusion coefficient (Crank, 1975).

For simplicity, the following dimensionless variables and parameters are used here:

0 = C/Co, r = tDo/R 2, X = r/R, ~ = VR/(CoDo)

km hR Km Co Sh Do

Equations (1) to (6) are then transformed into the following dimensionless form:

~?N=f(0) 0 ~ - + ~ ~ + f (0) ~ O+Km (7)

f ( 0 ) = 1 + 6 0 or exp(60) (8)

• =0; 0 = 0 (9)

subject to

~0 X = 0 ; ~ = 0 (10)

X = 1; f (0) 0~-= S h ( 1 - 0 ) . (11)

The function f ' (O) in (7) is the derivative of f (O) with respect to 0.

154 S .H . LIN

Equations (7) to (11) describe a highly nonlinear diffusion problem which is suitable for the orthogonal collocation method to be described below.

The orthogonal collocation method (Villadsen and Stewart, 1967) essentially involves approximation of the spatial derivative terms in the above equations by an orthogonal polynomial, which could be of the Jacobi, Legendre or the Chebychev type. The Legendre type has been shown by Villadsen and Stewart (1967) to be more suitable for the present problem because of its better convergence. According to this method, the dimensionless concentration is assumed to be approximated by the following expansion:

O(z,X)=O('c, 1)+ (1 -X 2) ~ aj('r )P j_ l (X 2) j = l

(12)

where aj(z) are the expansion coefficients, Pj_I(X 2) the orthogonal polynomial and n the order of the orthogonal approximation. Expressing in terms of the above expansion at the ith collocation point, (7) is transformed into

d0(i) ,+l ["+~i 12 c~0(i) (13) dz - f[O(i)] ~ Bi'j(j)+ f'[O(i)] Ai, j(j) O(i)+K m

j = l j

subject to

~=0; 0(i)=0 i = l , 2 , . . . , n + l . (14)

The boundary condition of (11) then becomes

F 1 f[O(n+l)J A,+I,~(j) =Sh[1-O(n+l)] Lj= i

(15)

which is a nonlinear algebraic equation for O(n+ 1). If O(n+ 1) obtained from (15)is substituted back into (13), there yields a set of ordinary differential equations which can be readily integrated by any stable numerical integration scheme, such as the fourth-order Runge-Kutta method used here (Lapidus, 1962). To obtain O(n+ 1) from the nonlinear algebraic equation (15), several iterative methods can be used. In this work, both the reguli-falsi position method and the Newton-Raphson method (Lapidus, 1962) were found to work equally well.

The accuracy of the orthogonal collocation method greatly depends on the order of orthogonal approximation, n. Villadsen and Stewart (1967)

N O N L I N E A R D I F F U S I O N I N B I O L O G I C A L S Y S T E M S 155

have shown that a third or higher order approximation yields very satisfactory results for general computat ional purposes. A sixth order approximation is used here. Crank (1975) has obtained an analytical solution for the simpler case without reaction, ~=0 , and with constant diffusion coefficient, 6 =0. This serves to check the accuracy of the present numerical solution. In fact, the analytical and the numerical solutions were found to be essentially identical, with a difference being less than 0.1 ~o which is very acceptable. The following figures show some typical results of steady and unsteady state concentration responses.

1.0

0.8

0,6

0.4

0.2

0 0,2 0.4 O. 6 0.8 1.0 X

Figure l. The effect of constant par~imeter 6 on the dimensionless concentration profiles without biological reaction. 6 =-0.5, - - - - 6

=0.0, - . . . . . . 6=0.5

3. Discussion of Results. Figure 1 shows the transient dimensionless concentration profiles for a linear concentration dependence of diffusion coefficient and without biological reaction. Both increase and decrease in diffusion coefficient with increasing dimensionless concentration are considered. It is apparent that the case with 6 = - 0 . 5 yields the steepest

156 S .H . LIN

dimensionless concentration profiles among the three cases under consideration. This is obviously due to strong retardation of dimensionless concentration diffusion at a reduced diffusion coefficient. For this parti- cular case without biological reaction, the final dimensionless concentration profiles for all 6 will be flat ones. However, the dimensionless time for the dimensionless concentration to reach its steady state varies markedly with

h e

0 ,8

0 . 6

0 , 4

0 . 2

I I I I

_ / - , / / . i i / , :__..--%.15 I / I

•

- / / / ! / _ 0 . 0 5 /

o.o

0 ,2 0 . 4 0 .6 0 . 8 1,0 X

Figure 2. The effect of the modified Sherwood number S h on the transient and steady state dimensionless concentration profiles with e=2.5 and f(0)= 1

+ 0.50. S h - - 1 . 5 , - - - - S h = 5.0, - . . . . . . S h = 50.0

the constant parameter, 6. For example, dimensionless times of 0.925, 0.575 and 0.441 are, respectively, required for the cases with 6 = - 0 . 5 , 0 and 0.5 to reach 99 ~o of the steady state.

The effect of the modified Sherwood number, S h , on the transient and steady state dimensionless concentration profiles is demonstrated in Figure 2. This figure clearly shows that as the modified Sherwood number increases, the dimensionless concentration diffusion is proportionally acce-

N O N L I N E A R D I F F U S I O N I N B I O L O G I C A L S Y S T E M S 157

lerated. Hence, the dimensionless concentration profile approaches its steady state faster as the modified Sherwood number increases. It is also observed that an increase in the modified Sherwood number increases the steady state dimensionless concentration profile. This is due to more solute diffused into the sphere at a reduced mass transfer resistance.

It is of interest to note in Figure 2 that as the modified Sherwood is ,:,increased beyond 50, increase in the dimensionless concentration profile becomes much less appreciable. For instance, the difference in the transient

i.o[ I T.~o I I

0.8

0.6

0.4

0.2

0 0 .2 0 .4 0 .6 0 .8 1.0 X

Figure 3. The effect Of the dimensionless maximum reaction rate on the transient and steady state dimensionless concentration profiles with Sh= 10

and f(O)+ 1 +0.50. e=0.5, - - - -c~=5.0, - . . . . . . c~= 10.0

or steady state dimensionless concentrations for S h = 5 0 and Sh= 100 1s about 0.4~o only and this figure will be further reduced as Sh increases beyond that. This indicates an important fact that for such a large modified Sherwood number, the whole diffusion process is controlled by the biological reaction because further increase in the modified Sherwood

158 S . H . L I N

number, which is equivalent to a decrease in mass transfer resistance, does not improve the dimensionless concentration diffusion. Under this condition, the diffusion is said to be biological reaction controlled and the dimensionless concentration profiles will become Very sensitive to the variation of the maximum reaction rate, e.

F i g u r e 3 displays the effect of the dimensionless maximum reaction rate, c~, on the dimensionless concentration responses. It appears that the dimensionless concentration diffusion is significantly retarded as the dirnen-

0 .8

1.0

0 ,6

0.4.

0.2

0 0.2 0.4 0.6 0.8 1.0 X

Figure 4. The effect of constant parameter 5 on the transient and steady state dimensionless concentration profile with nonlinear concentration dependence of diffusion coefficient and without biological reaction. - - - - - 6 =

-0.5, - - - -5 =0.0, - . . . . . . 5 =0.5

sionless maximum reaction rate increases. This is anticipated because part of the solute diffused into the sphere is consumed by the biological reaction. It is also observed that as the dimensionless maximum reaction rate increases, the dimensionless time for the dimensionless concentration to reach its steady state decreases. For instance, the dimensionless times

NONLINEAR D I F F U S I O N IN BIOLOGICAL SYSTEMS 159

required for the cases with e = 0 . 5 and e = 10 to reach 99 ~o of their s teady state are 0.477 and 0.321, respectively. This dimensionless t ime will be further reduced too as the dimensionless m a x i m u m react ion rate is increased beyond that .

The result cor responding to Figures 1, 2 and 3 for the case with nonl inear concent ra t ion dependence of diffusion coefficient, f (O)=exp(60) , are demons t ra t ed in Figures 4, 5 and 6. Compar i son of Figures 4, 5 and 6 with the corresponding one of the previous case indicates the similari ty in

1,0

0,~

0.6

0

0.4

0.2

0 0.2 0.4 0.6 0.8 l.O X

Figure 5. The effect of the modified Sherwood number Sh on the transient and steady state dimensionless concentration profiles with c~=2.5 and f(O)

=exp(0.50). Sh = 1.5, - - - - Sh = 5.0, - . . . . . . Sh = 50.0

general characteristics. The only difference between them is tha t for the same cons tant pa ramete r 6, the diffusion coefficient is more affected by the dimensionless concent ra t ion for the present case than for the previous one. Except for this point, observat ions made in Figures 1, 2 and 3 still hold true for the present case.

160 S .H. LIN

I'° I I I _ _ _ _ I T~IXI

0.8

0.6

t . . - - . 1 - - - - - - - ~

0 . 4

0 . 2

0 0,2 0.4 0.6 0.8 1,0 x

FiguYe~:i~i.'.':::).The effect of the dimensionless max imum react ion rate on the tra/asi~n~"::and steady state dimensionless concentration profiles with Sh= 10.0

and Jf (0) = exo(0.50)._., c~ = 0.5, - - - - c~ = 5.0, - . . . . . . c~ = 10.0

4. Conclusions. An analytical procedure is presented in this work for predicting the transient and steady state dimensionless concentration profiles of a diffusion system with variable diffusion coefficient and with biological reaction. The highly nonlinear governing diffusion equation is solved by the orthogonal collocation method which involves the transfor- mation of the partial differential equation into a set of ordinary differential equations. The ordinary differential equations in turn are integrated by a stable numerical integration scheme. Because of high accuracy and computat ional stability of this method, very satisfactory results can be obtained with low order of orthogonal approximation. In fact, a sixth order approximation used here yields essentially identical solutions to the analytical ones.

Observations of the numerical results indicate that the dimensionless concentration profiles may become flatter or steeper depending on positive

NONLINEAR DIFFUSION IN BIOLOGICAL SYSTEMS 161

or negative effect of the dimensionless concentration on the diffusion coefficient. It is also found that the diffusion system exhibits the characteristics of biological reaction control at high modified Sherwood number greater than 50. Under this condition, the biological reaction becomes a dominating factor in the determination of the dimensionless concentration distribution in the sphere.

LITERATURE

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Math. Biol., 35, 59-67• Herschkowitz-Kaufman, M. 1975. "Bifurcation Analysis of Nonlinear Reaction-Diffusion

Equations." Bull. Math. Biol., 37, 589-636. Lapidus, L. 1962. Digital Computation for Chemical Engineers. New York: McGraw-Hill. Lin, S. H. 1976. "Oxygen Diffusion in a Spherical Cell with Nonlinear Oxygen Uptake

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Diffusion Equations." Bull. Math. Biol., 39, 391-396• Vaccarezza, L. M., J. L. Lombardi and J. Chirife. 1974. "Kinetics of Moisture Movement

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Villadsen, J. W. and W. E. Stewart. 1967. "Solution of Boundary Value Problems by Orthogonal Collocation." Chem. Engng Sci., 22, 1438-1501.

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Nonlinear diffusion in biological systems