Comparison of the relative concentration of motile and non-motile bacteria in small pores

J. theor. Biol. (1988) 133, 499-512

Comparison of the Relative Concentration of Motile and Non-motile Bacteria in Small Pores CHRISTOPHER T. SKOWLUNDJ" AND DALE W. K1RMSE

Chemical Engineering Department, University of Florida, Gainesville, Florida 32611, U.S.A. (Received 17 February 1988, and in revised form 4 May 1988) The effect of small pores (similar in size to the stomata of plants) on the diffusion constants and relative concentrations of non-motile, randomly motile and chemotactic bacteria is considered. It is shown that although the Brownian diffusion constant of non-motile bacteria is a couple of orders of magnitude lower than the diffusion constant of motile bacteria, non-motile bacteria will still be present in both short (100o~m) and long (0.Scm) pores in similar numbers to motile bacteria. It is postulated that this is due, at least in part, to the smaller amount of excluded volume for non-flagellated bacteria. 1. Introduction

In the literature, m a n y examples o f theoretical advantages of motile or chemotactic bacteria over non-motile bacteria can be found (cf. Lauffenburger et al., 1981; Lauffenburger & Calcagno, 1983). However, little theoretical work has been done to c o m p a r e the relative advantages of the different modes o f diffusion in confined spaces or pores. Since this p h e n o m e n o n is important in the s t u d y of both p h y t o p a t h o g e n i c bacteria and anaerobic fermentation of plants in the rumen or digester, it is important to understand the possible differences between results for simple bulk diffusion. Experimentally, a n u m b e r of studies have c o m p a r e d the relative concentrations o f motile and non-motile bacteria found inside invaded or fermenting plants. In a study of p h y t o p a t h o g e n i c bacteria, Panopoulos & Schroth (1974) c o m p a r e d invasion by a motile species of bacteria with invasion by its non-motile (but flagellated) mutant. Their results showed that the motile strain and the non-motile mutant were present inside the plant at a ratio of a p p r o x i m a t e l y 10: 1. In contrast, a similar study by Diachun et al. (1944) using two different species of bacteria, one motile and one non-motile (but without flagella), reported equivalent concentrations o f motile and non-motile bacteria inside the plant. Similarly, Latham (1980) reported a ratio of only 2 : 1 in favor of the motile species, in a study of the n u m b e r of bacteria inside plant material undergoing digestion in the rumen. What we intend to show is that the similarity in the n u m b e r o f flagellated and non-flagellated bacteria found in these samples is due, at least in part, to the effects t Present address: Center for Biochemical Engineering Research, New Mexico State University, Las Cruces, N.M., 88003-001, U.S.A. 499 0022-5193/88/120499 + 14 $03.00/0 © 1988 Academic Press Limited

500

C.T. SKOWLOND AND I). W. KIRMSE

o f small pores on the diffusion constant and equilibrium concentration o f motile bacteria.

2. System Description First, we must describe what is meant by the term " p o r e " . Bacteria are unable to penetrate the intact surfaces of plants (Billing, 1982). It has been observed that bacteria will penetrate the internal structure of plants that have undergone some sort of pretreatment (such as chewing, grinding, chopping), via the lumen or intercellular spaces (Baker & Harriss, 1947; Monson et al., 1972). For plants that are undamaged, the stomata have been observed to be the entry point for bacteria (Cheng et al., 1980). In this paper the general term " p o r e " will refer to either of these entrances. Although it is difficult to determine an "average" pore size, medium-sized stomata are usually between 10 and 15 txm in diameter and up to 40 Ixm or more in length (Steward, 1959; Willmer, 1983). For intercellular spaces or lumens, similar diameters seem reasonable (cf. Soltes, 1983) although their lengths may be much longer. The system to be modeled will consist of a bulk fluid which contains the bacteria and plants. The pores will be assumed to be filled with water (allowing us to neglect the time necessary for the pores to fill) and free o f bacteria. Although there are a number o f mechanisms by which bacteria can be transported from the bulk to a solid/liquid interface (Characklis & Cooksey, 1983), we will consider only Brownian motion and motility/chemotaxis. Furthermore, since it has been observed that phytopathogenic bacteria can live in the intercellular spaces of plants without adsorbing onto the surface o f the plant cells (Sequeira et al., 1977; Huang & Van Dyke, 1978; Hildebrand et al., 1980), adsorption in the pores will be neglected.

3. Effect of Small Pore on Diffusion Constants For particles undergoing Brownian diffusion, the effect of small pores on the diffusion constant has been discussed at length elsewhere, and so will not be repeated here (cf. Anderson & Quinn, 1974; Brenner & Gaydos, 1977). It is sufficient to note that increased friction due to small pores can be substantial. For example, for a bacterial cell o f radius 0.5 I~m in a pore o f radius 5 t~m, the resulting Brownian diffusion constant will be 36% lower than the value in the bulk (Paine & Scherr, 1975). For motile bacteria, the effect of the confined space is not so straightforward. For example, experimental observations on the effect of increased fluid viscosity on motility have shown that diffusivity increases with increasing viscosity (Shoesmith, 1960; Schneider & Doetsch, 1974), which is contrary to what would be expected for other types of diffusion (e.g., Brownian). In addition to the effect o f friction, we must also take into consideration the effect of the confined space in relation to the mean distance travelled by motile bacteria during a run. For motile diffusion in the bulk, Lovely & Dahlquist (1975) showed

CONCENTRATION

OF

BACTERIA

1N S M A L L

PORES

501

that the diffusion constant can be calculated using the equation DM=

(1)

vL,,

3

where v = velocity of motile bacteria during a run and L,, = mean distance travelled during a run (i.e., the m e a n free path). Because of the assumptions used in deriving eqn (1), it is similar to the diffusion of a gas as predicted by the kinetic theory o f gases (Present, 1958). For a gas, the mean free path refers to the distance travelled by the gas molecule between collisions with other gas molecules. Implicit in both derivations is the assumption that the distance travelled is not affected by any extraneous factors (such as the pore wall). For diffusion in small pores this assumption is obviously not correct. I f the mean free path is much longer than the pore diameter, then most collisions will be with the pore wall. This type of diffusion is known as K n u d s o n Diffusion. I f we assume that diffusion in a small pore is similar to that of gases (i.e., most changes in direction are due to collisions with the pore wall), then the diffusion constant for motile bacteria will be given by the equation for K n u d s o n Diffusion (Present, 1958) 2vR DM

-

(2)

3

where R = radius of the pore. Equation (2) assumes that the bacteria do not adsorb onto the pore wall and that diffusion within the pore remains random. Using eqn (2), we see that the small pore can have a great effect on the bacterial diffusion constant. For example, using the case given by Lovely & Dahlquist (1975), the diffusion constant for the species they used was 1.3 x 10 -5 cm2/s. The measured velocity was 20 I~m/s, giving a mean free path o f 200 I~m. For a pore of radius R = 5 I~m, assuming K n u d s o n Diffusion, the pore diffusion constant would be only 6.7 x 10 -7 cm2/s, almost 1/20th o f the bulk diffusion value. 4. Diffusion into Short Pores Assuming a pore of length L and radius R, the dimensionless flux o f non-motile bacteria into the pore is (3)

JN = -SNa(ne-ns),

with the dimensionless variables given by J* L JN

-

D N

b ,

DN 8N -- D '

kcL cl = D '

N n

-- Nb"

(4)

Here J * = dimensional flux, D = diffusivity of glucose, N = concentration of nonmotile bacteria, DN = Brownian diffusion constant of the non-motile bacteria, kc = mass transfer coefficient. Subscripts e, s and b stand for equilibrium, pore mouth and bulk values, respectively.

502

c.T.

SKOWLUND

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KIRMSE

The Brownian diffusion constant is given by the Stokes-Einstein equation DN --

kT

(5)

6n-tza '

where k = Boltzmann's constant, T = temperature,/z = fluid viscosity, a = radius of bacterial cell. The mass transfer coefficient is given by the equation (Kelman, 1965) kc

4D rrR

or

4L a=rrR

(6)

Because of the finite size of the bacteria, the equilibrium bacterial concentration will be less than the bulk concentration. This is due to the excluded volume present for finite particles in small pores (Anderson & Quinn, 1974). This difference in concentrations is expressed in terms of a "distribution coefficient", which is defined as the ratio of the pore and bulk concentrations at equilibrium. If we assume that the bacterial cells are spherical and the pores cylindrical, then the distribution coefficient for the non-motile bacteria is given by the equation (Giddings et al., 1968) qbN=

1--

=he.

(7)

For randomly motile bacteria, an equation similar to equation (3) can be used for the dimensionless flux:

JM = - 8 , ~ ( ~ , , - m s ) .

(8)

The random motility diffusion constant ~M will be much larger than the Brownian diffusion constant 6N, allowing the motile bacteria to diffuse into the pores much quicker than the non-motile bacteria. However, due to the flagella of the motile bacteria, the distribution coefficient of the motile bacteria, qbM, will be smaller than that of the non-motile bacteria, ~N. Since flagella are rigid (Fujime et al., 1972), they will tend to exclude the bacteria from a larger portion of the pore volume. Because of the length of these flagella, on average between 5 and 10 I~m (Hazelbauer & Harayama, 1983), the effect on the distribution coefficient will be important. If we assume that flagella dimensions result in distribution coefficients similar to those for capsular particles (see Fig. 1), then the distribution coefficient of the motile bacteria can be calculated by the equation (Giddings et al., 1968) (9)

(~ M = (~ Nf~ F,

where ~ F is ~F = 1-~

h

(1 + h 2 ) E

,

for h < 1

(10)

forh>l.

(11)

and qbv=l_3___~

(l+h2) E

,

+(l_h2)F

rr, 1

CONCENTRATION

OF

BACTERIA

('~

IN

SMALL

PORES

503

Non- motile

Flogello

Motile

\

L~-

%2 i;22222222 i Sii211 ?! FIG. 1. Assumed shapes of non-motile and motile bacteria. Also shown: capsule approximation for the shape of motile bacteria (dash).

The functions F ( ~ , k) and E (~b, k) are elliptic integrals of the first and second kind, respectively, and

LF

h=a 2(R ~ -'

(12)

where L F is the length of the flagella (see Fig. 1). The effect of the flagella on the distribution coefficient can be quite substantial. For example, assuming a bacterial cell of radius a = 0.5 Ixm diffusing into a pore o f radius R = 5 ~m, we obtain a distribution coefficient of @N = 0.81 for non-motile bacteria (which do not have flagella). However, a motile cell with flagella of LF = 10 l~m has a distribution coefficient of only 0.093, almost one order of magnitude smaller. (Figure 2 gives a wider comparison of the distribution coefficients for motile and non-motile bacteria.)

1.00 0.80 0"60 4

0.40

i

=7-5/zm

0"20

5

to

15 L,,-(/.¢m)

20

25

FIG. 2. Effect of flagella on the distribution coefficient of the motile bacteria for three different pore

radii.

504

c . T . SKOWLUND AND D. W. KIRMSE

The importance of the difference in these distribution coefficients can easily be illustrated. Assume we have a pore of length L = 100 I~m. The concentration of non-motile bacteria in the pore can be given by the equation

On

02n

~=

8N ~-x2,

(13)

where x is the dimensionless position within the pore (x = 1 is the pore mouth) and t is the dimensionless time (=t*D/L2). The boundary conditions for this equation are

an t =0. OX x=0

O~nt =a(dPN-ns), OX x=l

(14)

The initial condition is simply n = 0 for 0 - x -< 1. We have a similar set of equations for motile bacteria. The analytical solution for eqn (13) is given elsewhere (Crank, 1975). A comparison of the average pore concentrations for motile and non-motile bacteria is given in Fig. 3. Note that the motile bacteria reach their equilibrium concentration (0.093) very quickly, at approximately five minutes, but by eight minutes the concentration of non-motile bacteria within the pore has surpassed that of the motile bacteria, even though a much longer time is necessary for equilibrium to occur. This is due to the difference in the respective distribution coefficients. The above example assumes that the motile bacteria diffuse into the pores randomly, but it has been observed that plants can have a chemotactic effect on the diffusion of bacteria into stomata (Chet et aL, 1973). Therefore, it is important to determine what effect chemotaxis may have on the diffusion of motile bacteria into the pores. 0.20

046

0.I 2

0-08

0.04

0

I

I

240

J

I

480

I

I

720 t*(sec)

i

t

960

I

1200

FIG. 3. Average pore concentration of non-motile (solid) and randomly motile (dash) bacteria in a pore of radius R = 5 v.m and length L = 100 tzm. (,5M/8N = 100; LF = 10 v.m; a = 0-5 v.m).

CONCENTRATION

OF

BACTERIA

IN

SMALL

PORES

505

The flux of chemotactic bacteria is given by the equation (Keller & Segel, 1971) Jc = --SM

_~_ m ag +8c ox g ax

(15)

where 8c = dimensionless diffusivity related to the chemotactic flux of cells (typical values of 8c are two to three times 8ta (Holz & C h e n , 1979)) and g -- the dimensionless concentration of chemotactic agent (assumed to be glucose). To solve for the pore concentration of bacteria, we must first derive a boundary condition for the pore mouth, x = 1. This can be accomplished using the "film theory" for diffusion between two phases (Bird et ai., 1960). Basically, this theory states that there is a finite distance outside the pore mouth through which the concentration of, for example, the glucose will change from gb to gs. If we assume that the distance over which the concentration of randomly motile bacteria changes from mh to m~ is equal to the distance required for gb to change to g~ (which is true if we use eqn (6) to calculate the film thickness), then we can assume that the film thickness of the chemotactic bacteria will equal that o f the chemotactic agent (g). Using this assumption, the flux at the pore mouth is given as Jc =

8Ma(1 - fl)(1 - S)(OM -- m,S/~) S - S~

(16)

where fl = 8c/8M and S-- gb/g~. (A more detailed derivation of eqn (16) is given in the Appendix.) Taking the limit of eqn (16) as S ~ 1, gives the equation for the randomly motile bacteria (eqn (8)). A comparison with eqn (8) also shows that in order for the chemotactic bacteria to diffuse into the pore (Jc < 0), the following condition must be met m~ <~PM Lg-hh.I "

(17)

For randomly motile bacteria, the condition is simply m,
at

= cSta -~x 2 - a C a x

-~x

"

(18)

In order to solve eqn (18), we need an equation for the pore concentration of chemotactic agent. Assuming that the plant cells surrounding the pore contain an infinite amount of the chemotactic agent, the rate at which the agent diffuses into the pore is given by the equation

2 kwL2 gg = R---D--(1 - g ) =p2(1 - g ) , where k,. is the mass transfer coefficient from the plant cells into the pore.

(19)

506

c.T.

SKOWLUND

AND

D. W.

KIRMSE

Using eqn (19) in the chemotactic agent mass balance gives

Og

~_p2(1 _ g )

02g

(20)

at - O x 2

with the boundary conditions -Og] = ot(gb--gs) OX x=l

and

-Og -[

= a ~ ( g - 1),

(21)

OX x=O

where a,. = kwL/D. The initial condition is g = 1 for 0 - x <- 1. For chemotactic bacteria, we combine eqns (15), (16) and (21) for the boundary condition at x = 1 O~-xm '

=a

[(1 - ~)(1s~S)S~

eft(S-l)

]

ms-a

( 1 - f l ) ( 1 - S)qbM

S_S~

(22)

Equations (15) and (21) can be combined for the boundary condition at x = 0 --

-

8X

m.

(23)

x=0

The numerical solution for the above set of equations was obtained using an orthogonal collocation approximation in the x-direction, and a Runge-Kutta integration in the time dimension, as described by Finlayson (1972). (Note: due to the inherent stiffness of the equations near t = O, the "constant" bulk concentration was instead set equal to gb ( t ) = gb + ( 1 -

gb ) e-2°'.

(24)

Use of eqn (24) removed stiffness from the numerical solutions but should not affect the results substantially.) For the case of aw = 0 (i.e., no addition of the chemotactic agent to the pore from the surrounding plant cells), the duration of the chemotactic gradient is too short for chemotaxis to benefit the motile bacteria, unless the bacteria become irreversibly attached to the pore walls (see Fig. 4). For aw > 0, chemotaxis can benefit the motile bacteria if aw is high enough to permit strong chemotactic agent gradients within the pore (Fig. 5). Since the steady-state profile of chemotactic agent, as given by eqn (20), is reached fairly quickly--in less than one minute for the three values of aw shown--the steady-state solution of equation (20) can provide insight into the benefit of chemotaxis. The solution to eqn (20) with Og/Ot = 0 is cosh

(px ) + ot~ sinh (px ) P

g=l-(1-gb)

(l+_.~)cosh(p)+(p+P)sinh(p,

(25)

CONCENTRATION

OF

BACTERIA

IN

SMALL

PORES

507

6.0

4.8

3"6

2.4

1.2

I

0

120

I

I

240

I

I

[--i-

360 /°(sec)

I

480

600

FIG. 4. Average pore concentration of chemotactic bacteria for ct.. = 0 . ( / 3 = 3 ; gb = 10-3; other parameters are the same as in Fig. 3).

30

24-aw

18

12

6

/~-d----~---~ 0

120

I 240

I

I 360

I

] 480

[ 600

t°(sec) FIG. 5. Average pore concentration of chemotactic bacteria for a.. > 0. are the same as in Figs 3 & 4).

(gb = 0'1;

other parameters

Setting x = 1 gives the pore mouth concentration of the chemotactic agent (g,). From eqn (25) we can see that for values of gb << 1, gs will be relatively independent of gb. However, the ratio g,/gb will be inversely proportional to changes in gb, making calculations of the pore concentration of chemotactic bacteria heavily dependent on the value of gb. Although the chemotactic agent concentration reaches the steady-state profile relatively quickly, Fig. 5 shows that it will take much longer for chemotactic bacteria to reach their equilibrium values. The equilibrium pore concentration of chemotactic

508

C. T. S K O W L U N D

AND

D. W. K I R M S E

TABLE 1

Steady-state concentrations gb

a,,,f ,:l

g,~

g.Jg~,

0-1

0"1 0.01 0.001

0-36 0-20 0.13

[g~lgh]3~M

3'6 2"0 1'3

4.2 0"74 0.20

m,,v~ 80"0 47"0 4'5

bacteria can be calculated by solving equation (18) with Ore~or = 0 to give re=m,

~

=qb M g

(26)

kg~J

where g is given by eqn (25) and ms by eqn (17). Table 1 shows the values o f g.~ for the values of aw and gb used in Fig. 5. Also shown are the equilibrium pore mouth (m,) and average pore concentrations o f chemotactic bacteria.

5. Diffusion into Long Pores The previous section dealt with the diffusion of bacteria into pores similar in length to the stomata o f plants, and due to the short time necessary for substantial diffusion to occur, bacterial growth was neglected. In this section we will consider the relative concentrations o f bacteria in pores of much longer length. The ideal system that is considered is a fermenter filled with chopped plants, the individual pieces of which will be termed pellets. These pellets are assumed to be completely porous (i.e., the pores will penetrate the full length o f the pellet) and uniform in size. The pores are assumed to be uniform in length and radius, and bacterial growth rates are assumed to follow Monod kinetics. Initially, it is assumed that the concentration o f glucose (assumed to be both substrate and chemotactic agent) is equal throughout the pore and bulk, i.e., the plant material has been submerged in the fluid long enough for all the chemotactic agent to diffuse out of the plant cells. At time t = 0, the bacteria are added to the bulk, and fermentation and diffusion into the pores commence. For the bacteria, the dimensionless mass balance is Ow

0t

0

wg Jw + kx - - ax ks + g

(27)

kdw,

where the general variable w stands for both the motile and non-motile bacteria. Jw is the dimensionless flux o f bacteria, which will be either Fickian (for the non-motile and randomly motile bacteria) or chemotactic (as given by eqn (15)). The dimensionless parameters are 12L 2 k~ = D " "

Ks k~=--, "

Go

Kd L2

(28)

kd = D

Here t2 = maximum specific growth rate, Ks = Monod half-velocity constant, Go = initial glucose concentration and Kd = cell decay coefficient.

CONCENTRATION OF BACTERIA 1N SMALL PORES

509

For the glucose we have the dimensionless equation Og O2g Ot-Ox 2

k~ wg Yk~+g

(29)

where Y = growth yield coefficient. For the bulk we also need mass balances. These will be similar to eqns (27) and (29) except that we drop the diffusion terms and include a term for the transport of each quantity into the pores a Wb

wbg b

--

- k~ - at k,+gb

Ogb 8t

--

kawb + ctAJw

kx wbgb Y ks+gb

(30)

ctA(gb --gs),

(31)

where A-

LAy

(32)

E

Here Ao = interfacial area between the pellet and bulk per unit volume and e = void fraction of the fermenter. Jw will be given by eqns (3), (8) and (16) for non-motile, randomly motile and chemotactic bacteria, respectively. The b o u n d a r y condition will be similar to eqn (14) (for the non-motile or randomly motile bacteria), eqn (21) with aw = 0 (for glucose) and eqns (22) and (23) with a,, = 0 (for the chemotactic bacteria). The values of the parameters are similar to those reported by Ghosh & Pohland (1974): kx = 13, k~ = 0-01, Y = 0 - 1 7 , ka =0"46 and A =0.33. The results of the calculations for each type of diffusion (done separately) are shown in Fig. 6. Again we see that chemotaxis is beneficial, but also that the pore 50

40-

...-~ ........ ........... c

30-

""..

20-

tO-

O

2-5

5

7.5

I0

12.5

t*{hr)

FiG. 6. Averagepore concentration of chemotactic(solid, c), randomlymotile (solid, r) and non-motile (dash) bacteria. Also shown: bulk concentration of bacteria (dot). (L=0.5 cm; 8M/8N = 100; /3 =3; a =0.5 p.m; R=5pm; LF= 101.tm).

510

c.T.

SKOWLUND

AND

D. W. K I R M S E

concentration of non-motile bacteria is not negligible; although fewer than randomly motile bacteria, due to the higher distribution coefficient, they are present in substantial numbers. Due to the slower rate of diffusion, though, non-motile bacteria are present only in the outer section of the pore (x > 0-9), whereas the motile bacteria are present throughout the complete length of the pore. 6. Discussion

In the introduction, we mentioned several investigations comparing the relative concentrations of motile and non-motile bacteria found inside plants; a comparison of motile and non-motile flagellated bacteria showed a concentration ratio of 10:1 in favor of motile bacteria, but a comparison of motile and non-flagellated bacteria showed equivalent concentrations. We feel that this discrepancy is due, at least in part, to the effect of the flagella on the distribution coefficient (i.e. equilibrium concentration) of the motile bacteria. Our calculations have shown that this difference between the distribution coefficients can allow non-motile bacteria to compensate for their much slower rate of diffusion by having a much higher equilibrium concentration within the pores. This is true for both short (100 ~.m) and long (0-5 cm) pores. On the other hand, we also show that chemotaxis can remove this advantage by increasing the equilibrium concentration of motile bacteria within the pore. In addition to lower equilibrium concentrations, we have shown that the small pores may also affect the diffusion of motile bacteria by changing the type of diffusion from random-walk (as it is in the bulk) to Knudson (where most of the changes in direction are due to collisions with the pore wall). The resulting diffusion constants can be much lower than the bulk values, though still a couple of orders of magnitude higher than Brownian diffusion. REFERENCES ANDERSON, J. L. & QUINN, J. A. (1974). Biophys..L 14, 130. BAKER, F. & HARRISS, S. T. (1947). Nutr. Abstr. Rev. 17, 3. BILLING, E. (1982). In: Bacteria and Plants (M. Rhodes-Roberts & F. A. Skinner, eds), p. 51. New York: Academic Press. BIRD, R. B., STEWART, W. E. & LIGHTFOOT, E. N. (1960). Transport Phenomena. New York: John Wiley & Sons. BRENNER, H. & GAYDOS, L. J. (1977). J. Colloid lmerf. Sci. 58, 312. CHARACKLIS, W. G. t~ COOKSEY, K. E. (1983). Adv. appl. Microbiol. 29, 93. CHENG, K. J., FAY, J. P., HOWARTH, R. E. & COSTERTON, J. W. (1980). Appl. & Environ. Microbiol. 40, 613. CHET, 1., ZILBERSTEIN, I. & HENIS, Y. (1973). Physiol. Plant Pathol. 3, 473. CRANK, J. (1975). The Mathematics o.fD(ffusion. London: Oxford University Press. DIACHUN, S., VALLEAU, W. D. & JOHNSON, E. M. (1944). Phytopathology 34, 250. FINLAYSON, B. A. (1972). The Method of Weighted Residuals and Variational Principles. New York: Academic Press. FUJIME, S., MARUYAMA, M. & ASAKURA, S. (1972). J. tool. Biol. 68, 348. GIDDINGS, J. G., KUCERA, E., RUSSELL, C. P. & MYERS, M. N. (1968). 3". phys. Chem. 72, 4397. GHOSH, S. &POHLAND, F. G. (1974). 3". War. Pollut. Control Fed. 46, 748. HAZELBAUER, G. L. & HARAYAMA, S. (1983). Int. Rev. Cytol. 81, 33. HILDEBRAND, D. C., ALOSI, M. C. & SCHROTH, M. N. (1980). Phytopathology 70, 98.

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HOLZ, M. & CHEN, S. H. (1979). Biophys. J. 26, 243. HUANG, J. S. & VAN DYKE, C. G. (1978). Physiol. Plant Pathol. 13, 65. KELLER, E. F. & SEGEL, L. A. (1971). J. theor. Biol. 30, 225. KELMAN, R. B. (1965). Bull. math. Biophys. 27, 57. LATHAM, M. J. (1980). In: Microbial Adhesion to Surfaces (R. C. W. Berkeley, J. M. Lynch, J. Melling, P. R. Rutter & B. Vincent, eds), p. 339. Chichester, England: Ellis Horwood Ltd. LAUFFENBURGER, D. t~: CALCAGNO, B. (1983). Biotechnol. Bioengng. 25, 2103. LAUFFENBURGER, D., ARIS, R. & KELLER, K. H. (1981). Microb. Ecol. 7, 207. LOVELY, P. S. & DAHHLQUIST, F. W. (1975). J. theor. Biol. 50, 477. MONSON, W. G., POWELL, J. B. & BURTON, G. W. (1972). Agron. J. 64, 231. PAINE, P. L. & SCHHERR, P. (1975). Biophys. J. 15, 1087. PANOPOULOS, N. J. & SCI-IROTH, M. N. (1974). Phytopathology 64, 1389. PRESENT, R. D. (1958). Kinetic Theory of Gases. New York: McGraw-Hill. SCHNEIDER, W. R. & DOETSCH, R. N. (1974). J. Bact. 117, 696. SEQUEIRA, L., GAARD, G. X, DE ZOETEN, G. A. (1977). Physiol. Plant Pathol. 10, 43. SHOESMITH, J. G. (1960). J. gen. Microbiol. 22, 528. SOLTES, E. J. (1983). In: Biomass Utilization (W. A. Cote, ed.), p. 271. New York: Plenum Press. STEWARD, F. C. (ed) (1959). Plant Pathology. Vol. II. New York: Academic Press. WILLMER, C. M. (1983). Stomata. London: Longman.

APPENDIX Before we derive eqn (16), let us derive eqn (8) for randomly motile bacteria, using the "film theory" for diffusion between two phases. If it is assumed that the film thickness is very small, then we can neglect changes in the flux within the film (i.e. it will be constant). For Fickian diffusion, the flux of bacteria is given by the equation dm JM = - - S M - dy'

(A1)

where we have introduced the new independent variable y = x - 1. For the film we have the b o u n d a r y conditions rely=o= ms

and

m l y = d = mb,

(A2)

where d = L.~t,,,/L(LaI,, = the film thickness) and, for simplicity, we have neglected the effect o f the small pore size on the equilibrium concentration (i.e. ~M = 1). Since we assume that JM is constant, eqn (A1) can be integrated to give the flux through the film as 8M JM -

d

(ms-ms).

(A3)

C o m p a r i n g eqn (A3) with eqn (8) shows that d = 1/a. A similar derivation for the chemotactic agent gives J c = - d (gb -- gs).

(A4)

For both the randomly motile bacteria and the chemotactic agent we see the film thickness is the same ( = d) and dependent solely on the geometry of the pore (see eqn (6)). If we assume that the film thickness is an actual physical region, then,