- Email: [email protected]
Behavior of microorganisms and internal state variables
J. theor. Biol. (1981) 92,359-362
Behavior of Microorganisms Internal State Variables I. RICHARD
Department
and
LAPIDUS
of Physics and Engineering
Physics, Stevens Institute
of Technology, Hoboken, New Jersey 07030, U.S.A. (Received 2 June 1980, and in revised form 20 October 1980) It is shown that the dependence of cell density on the swimming parameters of microorganisms-the average speed, the average turning frequency and the motility--obtained by Oosawa and Nakaoka using a theory with internal state variables to describe the behavior of individual cells, may be derived using only macroscopic quantities. These results are compared with the predictions of a model in which a partial differential equation is used to describe the behavior of microbial populations.
Recently Oosawa & Nakaoka (1977) developed a theoretical description of the behavior of populations of microorganisms in which individual cells are treated as particles with internal state variables. However, the results which they obtained for the dependence of the density distribution of the cells on the swimming parameters-the average speed, the average turning frequency and the motility-do not depend upon the internal state variables. In fact their results are actually obtained by averaging over the internal state variables. The purpose of this note is to show that the results of Oosawa and Nakaoka can be obtained directly without introducing internal variables. Comparison is also made with a model using a partial differential equation to describe the behavior of a microbial population. Consider a population of microorganisms moving in one dimension. (Only one-dimensional motion is considered here merely to simplify the derivation of the results. The generalization to three dimensions is straightforward.) Let N+(x, t) and N-(x, t) be the density of cells moving to the right and left respectively at a point x and time t. The average’ speeds of individual cells to the right and to the left are given by v+(x, t) and v-(x, f) respectively, and the average turning frequencies for cells moving to the right and left are f+(x, t) and f-(x, t) respectively. 359
0022-5193/81/200359+04$02.00/0
@ 1981 Academic
Press Inc. (London)
Ltd.
360
1. R.
One may then write densities as
down
LAPIDUS
directly
the differential
aN+/at = -a(N+v+)/ax
and subtracting a(N+ +N-)/at
equations
= -a(N+v+
for the
+ f N - f+N,,
aN~/at=a(Nmvm)/ax+f+N+-f-N Adding
equations
.
(1) (2)
(1) and (2) one obtains -N
L’ ilirx-.
(3)
At equilibrium JN, /at = 0 and JN /at = 0. Then equation (3) becomes ~(N+L~+-N-L~
)/dx = 0,
15)
or N+v+-N-X
=constant.
But at the walls of the container
N+L’+ -N
N+a+=N
16)
c = 0. Thus,
1%
(7)
This is equation (22) obtained by Oosawa and Nakaoka, which has been obtained here without the use of any internal variables. Substituting equation (7) into equation (4) one obtains at equilibrium a(N+v+)/ax+f+N+-(f
L.-/L! )N+=o.
(8)
--t /r~)=0
(9)
or aln(N+v+)/ax+(f+/r,-
Equation (9) is identical with equation (23) obtained by Oosawa and Nakaoka. Equation (9) relates the cell density to the swimming parameters. In order to interpret equation (9) it is useful to define the mean free path of cells traveling to the right and to the left by d, = LT+/f+ and dp = v-/f-. Then equation (9) may be rewritten as
a In (N+v+)/ax
+(1/d,
- l/d -) = 0.
If d, = d- the second term vanishes and one obtains (using equation the result N = constant/
L‘.
(10) (6)) (11)
for all cells moving to the right or the left. This is equation (41) of Oosawa and Nakaoka. It states that the cell density varies inversely with the average speed if the mean free path is independent of direction. It is not necessary that v+ = v and f+ = fm separately.
BEHAVIOR
OF
361
MICROORGANISMS
It is interesting to note that if the mean free path is a constant one may rewrite equation (12) as N = constantldv. The motility
of the population
(12)
is given by
p=v2/f
=dv.
(13)
N = constant/p.
(14)
Thus,
Equation (14) is actually a general result which is also obtained by solving the “diffusion” equation (Lapidus & Levandowsky, 1981; Lapidus, 1981)
aN/at = a2(pN)/ax2.
(15)
At equilibrium equation (14) is a solution of equation (15). Equation (14) also follows from equation (10) in the general case when d, # d-. This may be seen as follows. The mean free path to the right and left must be proportional to the average mean free path. At any point in space the mean free paths may be approximated as l/d,=(l/d)[1*(1/2)ad/ax].
(16)
Then l/d+Substituting
l/d-
into equation
= (l/d)(ad/ax)
= a(ln d)/ax.
(17)
(10) a In (dvN)/ax
= 0,
(18)
= constant/p.
(19)
or N = constantldv
It is of interest to note that the result given in equation (14) is quite reasonable. It states that cells will accumulate where the motility is low. Normally the motility will depend upon the concentration of nutrients such that in regions of high concentration of nutrient the motility will be small. Thus, the cells will accumulate where the concentration of nutrient is high. A detailed dynamic description of this process is discussed elsewhere (Lapidus, 1981). The notion of internal variables to describe chemotactic behavior of bacteria and other microorganisms has been employed, for example, by Koshland and others (Koshland, 1977; Macnab & Koshland, 1972) to
362
I.
R.
LAPIL)US
describe a number of experiments on bacterial chemotaxis. However, it should be noted that the behavior of bacterial populations as observed experimentally can be accounted for by macroscopic phenomenological theory as noted by several authors (Lapidus & Schiller, 1974, 1976, 1978; Nossal, 1972; Nossal & Weiss, 1973; Rosen, 1973, 1975; Scribner, Segel & Rogers, 1974; Segel & Jackson, 1973). I would like to thank Dr. M Levandowsky for many useful discussions regarding mathematical models to describe the motion of microbial populations. REFERENCES KOSHLANL). LAPInUS, LAPIDUS.
D. E.. JR. / 19771. Science 196. 1055. I. R. 11981 I. J. rheor. Biol. 92, 345. 1. R. & LEVANDOWSKY, M. ( 198 1). In Biochemistry
and Physiology
of Proro~ou.
Vol. 4, p. 235. IM. Levandowsky and S. H. Hurrier, eds). New York: Academic Press. LAPIDUS, I. R. & SCHILLER, R. (1974). Biophyr. J. 14,825. LAPIDUS. I. R. & SCHIL.L.ER. R. (1976). Biophyv. J. 16. 779. LAPIDUS. I. R. & SCHII.I.ER, R. (1978). Biophys. J. 22, 1. MACNAB, R. & KOSHLAND, D. E., JR (1972). Proc. mm. Acad. SC;.. U.S.A. 69, 2509. NOSSAL, R. (1972). Math. Biosci. 13, 397. NOSSAL, R. & WEISS, G. H. (1973). J. theor. Biol. 41, 143. OOSAWA, F. & NAKAOKA. Y. (1977). J. theor. Riol. 66, 747. ROSEN, G. I 1973). J. Theor. Biol. 41, 201. ROSEN. G. (1975). Math. Biosci. 24, 17. SCRIBNER. T. L.. SEGEI , L. A. & ROGERS, E. H. I I Y741. J. rheor. Biol. 46, 189. SEGEI., L. A. & JACKSON, J. L. 11973). J. Mechnmrciwm. CeI/ Mob/it\ 2, 22.
Behavior of Microorganisms Internal State Variables I. RICHARD
Department
and
LAPIDUS
of Physics and Engineering
Physics, Stevens Institute
of Technology, Hoboken, New Jersey 07030, U.S.A. (Received 2 June 1980, and in revised form 20 October 1980) It is shown that the dependence of cell density on the swimming parameters of microorganisms-the average speed, the average turning frequency and the motility--obtained by Oosawa and Nakaoka using a theory with internal state variables to describe the behavior of individual cells, may be derived using only macroscopic quantities. These results are compared with the predictions of a model in which a partial differential equation is used to describe the behavior of microbial populations.
Recently Oosawa & Nakaoka (1977) developed a theoretical description of the behavior of populations of microorganisms in which individual cells are treated as particles with internal state variables. However, the results which they obtained for the dependence of the density distribution of the cells on the swimming parameters-the average speed, the average turning frequency and the motility-do not depend upon the internal state variables. In fact their results are actually obtained by averaging over the internal state variables. The purpose of this note is to show that the results of Oosawa and Nakaoka can be obtained directly without introducing internal variables. Comparison is also made with a model using a partial differential equation to describe the behavior of a microbial population. Consider a population of microorganisms moving in one dimension. (Only one-dimensional motion is considered here merely to simplify the derivation of the results. The generalization to three dimensions is straightforward.) Let N+(x, t) and N-(x, t) be the density of cells moving to the right and left respectively at a point x and time t. The average’ speeds of individual cells to the right and to the left are given by v+(x, t) and v-(x, f) respectively, and the average turning frequencies for cells moving to the right and left are f+(x, t) and f-(x, t) respectively. 359
0022-5193/81/200359+04$02.00/0
@ 1981 Academic
Press Inc. (London)
Ltd.
360
1. R.
One may then write densities as
down
LAPIDUS
directly
the differential
aN+/at = -a(N+v+)/ax
and subtracting a(N+ +N-)/at
equations
= -a(N+v+
for the
+ f N - f+N,,
aN~/at=a(Nmvm)/ax+f+N+-f-N Adding
equations
.
(1) (2)
(1) and (2) one obtains -N
L’ ilirx-.
(3)
At equilibrium JN, /at = 0 and JN /at = 0. Then equation (3) becomes ~(N+L~+-N-L~
)/dx = 0,
15)
or N+v+-N-X
=constant.
But at the walls of the container
N+L’+ -N
N+a+=N
16)
c = 0. Thus,
1%
(7)
This is equation (22) obtained by Oosawa and Nakaoka, which has been obtained here without the use of any internal variables. Substituting equation (7) into equation (4) one obtains at equilibrium a(N+v+)/ax+f+N+-(f
L.-/L! )N+=o.
(8)
--t /r~)=0
(9)
or aln(N+v+)/ax+(f+/r,-
Equation (9) is identical with equation (23) obtained by Oosawa and Nakaoka. Equation (9) relates the cell density to the swimming parameters. In order to interpret equation (9) it is useful to define the mean free path of cells traveling to the right and to the left by d, = LT+/f+ and dp = v-/f-. Then equation (9) may be rewritten as
a In (N+v+)/ax
+(1/d,
- l/d -) = 0.
If d, = d- the second term vanishes and one obtains (using equation the result N = constant/
L‘.
(10) (6)) (11)
for all cells moving to the right or the left. This is equation (41) of Oosawa and Nakaoka. It states that the cell density varies inversely with the average speed if the mean free path is independent of direction. It is not necessary that v+ = v and f+ = fm separately.
BEHAVIOR
OF
361
MICROORGANISMS
It is interesting to note that if the mean free path is a constant one may rewrite equation (12) as N = constantldv. The motility
of the population
(12)
is given by
p=v2/f
=dv.
(13)
N = constant/p.
(14)
Thus,
Equation (14) is actually a general result which is also obtained by solving the “diffusion” equation (Lapidus & Levandowsky, 1981; Lapidus, 1981)
aN/at = a2(pN)/ax2.
(15)
At equilibrium equation (14) is a solution of equation (15). Equation (14) also follows from equation (10) in the general case when d, # d-. This may be seen as follows. The mean free path to the right and left must be proportional to the average mean free path. At any point in space the mean free paths may be approximated as l/d,=(l/d)[1*(1/2)ad/ax].
(16)
Then l/d+Substituting
l/d-
into equation
= (l/d)(ad/ax)
= a(ln d)/ax.
(17)
(10) a In (dvN)/ax
= 0,
(18)
= constant/p.
(19)
or N = constantldv
It is of interest to note that the result given in equation (14) is quite reasonable. It states that cells will accumulate where the motility is low. Normally the motility will depend upon the concentration of nutrients such that in regions of high concentration of nutrient the motility will be small. Thus, the cells will accumulate where the concentration of nutrient is high. A detailed dynamic description of this process is discussed elsewhere (Lapidus, 1981). The notion of internal variables to describe chemotactic behavior of bacteria and other microorganisms has been employed, for example, by Koshland and others (Koshland, 1977; Macnab & Koshland, 1972) to
362
I.
R.
LAPIL)US
describe a number of experiments on bacterial chemotaxis. However, it should be noted that the behavior of bacterial populations as observed experimentally can be accounted for by macroscopic phenomenological theory as noted by several authors (Lapidus & Schiller, 1974, 1976, 1978; Nossal, 1972; Nossal & Weiss, 1973; Rosen, 1973, 1975; Scribner, Segel & Rogers, 1974; Segel & Jackson, 1973). I would like to thank Dr. M Levandowsky for many useful discussions regarding mathematical models to describe the motion of microbial populations. REFERENCES KOSHLANL). LAPInUS, LAPIDUS.
D. E.. JR. / 19771. Science 196. 1055. I. R. 11981 I. J. rheor. Biol. 92, 345. 1. R. & LEVANDOWSKY, M. ( 198 1). In Biochemistry
and Physiology
of Proro~ou.
Vol. 4, p. 235. IM. Levandowsky and S. H. Hurrier, eds). New York: Academic Press. LAPIDUS, I. R. & SCHILLER, R. (1974). Biophyr. J. 14,825. LAPIDUS. I. R. & SCHIL.L.ER. R. (1976). Biophyv. J. 16. 779. LAPIDUS. I. R. & SCHII.I.ER, R. (1978). Biophys. J. 22, 1. MACNAB, R. & KOSHLAND, D. E., JR (1972). Proc. mm. Acad. SC;.. U.S.A. 69, 2509. NOSSAL, R. (1972). Math. Biosci. 13, 397. NOSSAL, R. & WEISS, G. H. (1973). J. theor. Biol. 41, 143. OOSAWA, F. & NAKAOKA. Y. (1977). J. theor. Riol. 66, 747. ROSEN, G. I 1973). J. Theor. Biol. 41, 201. ROSEN. G. (1975). Math. Biosci. 24, 17. SCRIBNER. T. L.. SEGEI , L. A. & ROGERS, E. H. I I Y741. J. rheor. Biol. 46, 189. SEGEI., L. A. & JACKSON, J. L. 11973). J. Mechnmrciwm. CeI/ Mob/it\ 2, 22.