Vol.40, pp. 671 674 PergamonPressLtd. 1978.Printedin Great Britain © Socielyfor MathcmaticalBiology
Bulletin of Mathen~urlcul Biolog),
0007-4985/78/0901-0671S02.000
NOTE STEADY-STATE DISTRIBUTION OF BACTERIA CHEMOTACTIC TOWARD OXYGEN
•
GERALD ROSEN
Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A.
The phenomenological theory for the chemotaxis and c o n s u m p t i o n of oxygen by motile aerobic bacteria is shown to yield a remarkably simple one-dimensional steady-state solution for a congregation of bacteria close to the surface of an oxygen-depleted aqueous medium.
1. Introduction. It has been known for many years that certain species of motile bacteria exhibit chemotactic response toward oxygen, minerals, and various organic substances (Adler, 1966, 1969; Adler and Dahl, 1967). Thus, for example, aerobic peritrichous bacteria such as Escherichia coli swim with a stochastic motion biased in the direction of increasing oxygen concentration in an aqueous medium that contains less oxygen than other vital substances. A phenomenological diffusion theory governs the distribution of chemotactic bacteria cells along with the spatially-varying concentration of chemotactic agent, but the essential nonlinearity of the equations in this theory has precluded rigorous analytical solutions (Keller and Segel, 1971; Segel and Jackson, 1973; Rosen and Baloga, 1975, 1976; Rosen, 1975a, b, 1976). The present communication reports a remarkably simple one-dimensional steadystate solution for the distribution of bacteria chemotactic toward oxygen. Since this exact solution exhibits an analytical dependence on the distance coordinate and on the constant parameters of the theory, the solution is especially noteworthy for it affords a means of checking the details of the theory against future experimental observations. 671
672
GERALD ROSEN
2. Governing Nonlinear Diffusion Equations. Assuming that all motile cells in the population are equivalent in their random and chemotactic swimming and that the total number of cells is constant, we have the cell number conservation equation 0n --
?t
=
V
• (~Vn
-
6ns- 1Vs)
(1)
where n = n ( x , t) denotes the number per unit volume of bacteria cells in the neighborhood of the point x at time t, s = s ( x , t) denotes the oxygen concentration, and l~, 6 are the mobility and chemotactic flux coefficient, respectively. If the latter quantities are treated as constants (i.e., independent of n, s or x, t) in accordwith previous studies (Keller and Segel, 1971 ; Rosen and Baloga, 1975, 1976; Rosen, 1975a, b and works cited therein), then (1) becomes ~n
~- =/iV 2 n - b V " (ns- 1Vs).
(2)
For motile species of Escherichia coli, we have the approximate experimental estimates/~ & 0.25 cmZ/hr and 6 & 0.75 cmZ/hr (Keller and Segel, 1971, pp. 243 244). Letting ~ denote the fractional rate of 02 consumptio n per unit concentration of bacteria cells, the oxygen diffusion consumption equation takes the form - - = D V 2s-c~ns ~t
(3)
in which the oxygen diffusivity constant D ~0.065 cm2/hr at temperatures of biological interest. The value of the constant ~, dependent on the concentration of nutrients, etc., can be estimated for E. coli under typical experimental conditions as ~ &2.5 x 10-Scm3/cell-hr (Keller and Segel, 1971, pp. 242-243).
3. Steady-State Solution for Bacteria Close to the Surface of an OxygenDepleted Medium. Let us consider the one-dimensional steady-state solution to (2) and (3) with z denoting distance into an oxygen-depleted aqueous medium from the z = 0 surface, where the number per unit volume of bacteria cells and the oxygen concentration have the prescribed values n=no~ S
So J
at
z=0.
(4)
STEADY-STATE"DISTRIBUTION OF BACTERIACHEMOTACTIC
673
Under the usual experimental conditions, no might typically be of the order 108 cells/cm 3, while So must be less than the solubility value of 02 at the medium temperature (e.g., 48.0mgOz/liter at 15~C, decreasing to 35.9mg Oz/liter at 3 0 C for an aqueous medium with an atmosphere of pure oxygen above the surface, and approximately 20'),o of these solubility values for an aqueous medium with fresh air above the surface). Suppose that oxygen and bacteria cells are absent far below the surface, so that n-~}s
at
z=,:~.
(5)
Specializing (2) and (3) for n=n(z) and s=s(z), we have
~ d~z2 - - 8
f/s
1
(6)
=0
dZs
D dz 2 - ~ns = 0.
(7)
The solution to (6) and (7) subject to the boundary conditions (4) and (5) is given by the remarkably simple closed-form expressions n=no[l+(z/z,)]
2
(8)
s =SoD + (z/z,)] -2"~-'
(9)
in which the characteristic thickness of the bacterial congregation and oxygencontaining layer appears as z, = [ (D~- in o 1)(2/~a- 1 )( 1 + 2l~6- 1)]1/2.
(10)
Note that the thickness formula (10) is wholly independent of So and depends on the motility and chemotactic flux coefficient exclusively through the dimensionless ratio #6-1. The total number of bacteria cells under a unit surface area follows from the solution function (8) as
~.U=
n(z)dz=n o o
[l+(z/z,)] o
2dz=noz,,
(11)
674
GERALD ROSEN
and hence by using formula (10) we obtain ....*'z, = D e - 12p6 - 1 (1 + 2p6 - 1).
(12j
The result (12) shows that the thickness z, varies inversely with the total number of bacteria cells under a unit surface area, z, oc~4..... 1, if conditions influencing the values of ~ and /~b-1 are held fixed. F o r the experimental parameter values cited above for E. coli, formula (12) yields ~ + ' z , & 3 x 106cells/cm, a relation that can be compared with future experimental measurements. This work was supported by NASA grant N S G 3090.
LITERATURE Adler, J. 1966. "Chemotaxis in Bacteria." Science, 153, 708 715. . 1969. "Chemoreceptors in Bacteria." Science, 166, 1588-1597. --and M. M. Dahl. 1967. ~A Method for Measuring the Motility of Bacteria and for Comparing Random and Non-Random Motility." J. Gen. Microbiol., 46, 161-173. Keller, E. F. and L. A. Segel. 1971. "Travelling Bands of Chemotactic Bacteria: A Theoretical Analysis." J. Theor. Biol., 30, 235 248. Rosen, G. 1975a. "Bacterial Chemotaxis in the Temporal Gradient Apparatus." Math. Biosci., 24, 17 20. . lC)75b. "Analytical Solution to the Initial-Value Problem for Travelling Bands of Chemotactic Bacteria." J. Theor. Biol., 49, 311-321. . 1976. "'Existence and Nature of Band Solutions to Generic Chemotactic Transport Equations." J. Theor. Biol., 59, 243-246. and S. Baloga, 1975. "'On the Stability of Steadily Propagating Bands of Chemotactic Bacteria." Math. Biosci., 24, 273-279. . 1976. ~On the Structure of Steadily Propagating Rings of Chemotactic Bacteria." J. Mechanochem. Cell Motility, 3, 225-228. Segel, L. A. and J. L. Jackson. 1973. "Theoretical Analysis of Chemotactic Movement in Bacteria." J. Mechanochem. Cell Motility, 2, 25 34.
Bulletin of Mathen~urlcul Biolog),
0007-4985/78/0901-0671S02.000
NOTE STEADY-STATE DISTRIBUTION OF BACTERIA CHEMOTACTIC TOWARD OXYGEN
•
GERALD ROSEN
Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A.
The phenomenological theory for the chemotaxis and c o n s u m p t i o n of oxygen by motile aerobic bacteria is shown to yield a remarkably simple one-dimensional steady-state solution for a congregation of bacteria close to the surface of an oxygen-depleted aqueous medium.
1. Introduction. It has been known for many years that certain species of motile bacteria exhibit chemotactic response toward oxygen, minerals, and various organic substances (Adler, 1966, 1969; Adler and Dahl, 1967). Thus, for example, aerobic peritrichous bacteria such as Escherichia coli swim with a stochastic motion biased in the direction of increasing oxygen concentration in an aqueous medium that contains less oxygen than other vital substances. A phenomenological diffusion theory governs the distribution of chemotactic bacteria cells along with the spatially-varying concentration of chemotactic agent, but the essential nonlinearity of the equations in this theory has precluded rigorous analytical solutions (Keller and Segel, 1971; Segel and Jackson, 1973; Rosen and Baloga, 1975, 1976; Rosen, 1975a, b, 1976). The present communication reports a remarkably simple one-dimensional steadystate solution for the distribution of bacteria chemotactic toward oxygen. Since this exact solution exhibits an analytical dependence on the distance coordinate and on the constant parameters of the theory, the solution is especially noteworthy for it affords a means of checking the details of the theory against future experimental observations. 671
672
GERALD ROSEN
2. Governing Nonlinear Diffusion Equations. Assuming that all motile cells in the population are equivalent in their random and chemotactic swimming and that the total number of cells is constant, we have the cell number conservation equation 0n --
?t
=
V
• (~Vn
-
6ns- 1Vs)
(1)
where n = n ( x , t) denotes the number per unit volume of bacteria cells in the neighborhood of the point x at time t, s = s ( x , t) denotes the oxygen concentration, and l~, 6 are the mobility and chemotactic flux coefficient, respectively. If the latter quantities are treated as constants (i.e., independent of n, s or x, t) in accordwith previous studies (Keller and Segel, 1971 ; Rosen and Baloga, 1975, 1976; Rosen, 1975a, b and works cited therein), then (1) becomes ~n
~- =/iV 2 n - b V " (ns- 1Vs).
(2)
For motile species of Escherichia coli, we have the approximate experimental estimates/~ & 0.25 cmZ/hr and 6 & 0.75 cmZ/hr (Keller and Segel, 1971, pp. 243 244). Letting ~ denote the fractional rate of 02 consumptio n per unit concentration of bacteria cells, the oxygen diffusion consumption equation takes the form - - = D V 2s-c~ns ~t
(3)
in which the oxygen diffusivity constant D ~0.065 cm2/hr at temperatures of biological interest. The value of the constant ~, dependent on the concentration of nutrients, etc., can be estimated for E. coli under typical experimental conditions as ~ &2.5 x 10-Scm3/cell-hr (Keller and Segel, 1971, pp. 242-243).
3. Steady-State Solution for Bacteria Close to the Surface of an OxygenDepleted Medium. Let us consider the one-dimensional steady-state solution to (2) and (3) with z denoting distance into an oxygen-depleted aqueous medium from the z = 0 surface, where the number per unit volume of bacteria cells and the oxygen concentration have the prescribed values n=no~ S
So J
at
z=0.
(4)
STEADY-STATE"DISTRIBUTION OF BACTERIACHEMOTACTIC
673
Under the usual experimental conditions, no might typically be of the order 108 cells/cm 3, while So must be less than the solubility value of 02 at the medium temperature (e.g., 48.0mgOz/liter at 15~C, decreasing to 35.9mg Oz/liter at 3 0 C for an aqueous medium with an atmosphere of pure oxygen above the surface, and approximately 20'),o of these solubility values for an aqueous medium with fresh air above the surface). Suppose that oxygen and bacteria cells are absent far below the surface, so that n-~}s
at
z=,:~.
(5)
Specializing (2) and (3) for n=n(z) and s=s(z), we have
~ d~z2 - - 8
f/s
1
(6)
=0
dZs
D dz 2 - ~ns = 0.
(7)
The solution to (6) and (7) subject to the boundary conditions (4) and (5) is given by the remarkably simple closed-form expressions n=no[l+(z/z,)]
2
(8)
s =SoD + (z/z,)] -2"~-'
(9)
in which the characteristic thickness of the bacterial congregation and oxygencontaining layer appears as z, = [ (D~- in o 1)(2/~a- 1 )( 1 + 2l~6- 1)]1/2.
(10)
Note that the thickness formula (10) is wholly independent of So and depends on the motility and chemotactic flux coefficient exclusively through the dimensionless ratio #6-1. The total number of bacteria cells under a unit surface area follows from the solution function (8) as
~.U=
n(z)dz=n o o
[l+(z/z,)] o
2dz=noz,,
(11)
674
GERALD ROSEN
and hence by using formula (10) we obtain ....*'z, = D e - 12p6 - 1 (1 + 2p6 - 1).
(12j
The result (12) shows that the thickness z, varies inversely with the total number of bacteria cells under a unit surface area, z, oc~4..... 1, if conditions influencing the values of ~ and /~b-1 are held fixed. F o r the experimental parameter values cited above for E. coli, formula (12) yields ~ + ' z , & 3 x 106cells/cm, a relation that can be compared with future experimental measurements. This work was supported by NASA grant N S G 3090.
LITERATURE Adler, J. 1966. "Chemotaxis in Bacteria." Science, 153, 708 715. . 1969. "Chemoreceptors in Bacteria." Science, 166, 1588-1597. --and M. M. Dahl. 1967. ~A Method for Measuring the Motility of Bacteria and for Comparing Random and Non-Random Motility." J. Gen. Microbiol., 46, 161-173. Keller, E. F. and L. A. Segel. 1971. "Travelling Bands of Chemotactic Bacteria: A Theoretical Analysis." J. Theor. Biol., 30, 235 248. Rosen, G. 1975a. "Bacterial Chemotaxis in the Temporal Gradient Apparatus." Math. Biosci., 24, 17 20. . lC)75b. "Analytical Solution to the Initial-Value Problem for Travelling Bands of Chemotactic Bacteria." J. Theor. Biol., 49, 311-321. . 1976. "'Existence and Nature of Band Solutions to Generic Chemotactic Transport Equations." J. Theor. Biol., 59, 243-246. and S. Baloga, 1975. "'On the Stability of Steadily Propagating Bands of Chemotactic Bacteria." Math. Biosci., 24, 273-279. . 1976. ~On the Structure of Steadily Propagating Rings of Chemotactic Bacteria." J. Mechanochem. Cell Motility, 3, 225-228. Segel, L. A. and J. L. Jackson. 1973. "Theoretical Analysis of Chemotactic Movement in Bacteria." J. Mechanochem. Cell Motility, 2, 25 34.
