- Email: [email protected]
Analysis of bacterial Chemotaxis in flowing water
Analysis of Bacterial Chemotoxis
in Flowing Water
I. RICHARD LAPIDUS Depariment of Physics, Stew-~ Instiiuie of Technology, Hobokeq
New Jersey 07030
Receiced 27 January 1980
ABSTRACT Chemotactic behavior by bacterial populations is studied using a mathematical model to analyze the response of swimming bacteria to a pulse of chemotactic attractant injected into flowing water. Experiments by Walsh and Mitchell are discussed using this model, and it is shown the experimental results can be understood in terms of the predictions of the model. The observations are consistent with biased random swimming by the bacteria. The model also makes predictions of the results of new experiments which are suggested for future research.
INTRODUCTION Chemotaxis by bacteria has been investigated in a number of different organisms, including Escherichia coli [l, 21, Salmonella typhimurium [4], Bacillus subtilis [5], Bdellovibrio bacteriovorus [27], Myxococcus xanthus [ 161, and others [26]. The experimental techniques have included the Adler [1] assay, the migration of bacterial populations in fixed attractant concentration gradients observed by Dahlquist et al. [4], and the formation and propagation of bands and rings which are generated by the consumption of attractant to maintain a moving chemical gradient [ 11. Recently
MA
TICAL
I. RICHARD LAPIDUS
80
increasing flow rate, and the time to reach this maximum value also decreased approximately linearly. These results may be understood qualitatively by noting that if the flow rate is low, the time for the bacteria to accumulate near the pulse of attractant will be large, and this will give rise to a large peak as the attractant is slowly washed out. On the other hand, if the flow rate is large, the time for the bacteria to accumulate before washout occurs will be short and the peak size will necessarily be small. Despite the fact that it is possible to understand the experimental results obtained by W & M qualitatively, it is of interest to obtain quantitative agreement between a theoretical formulation capable of making numerical predictions and the experimental data. It will be shown that the experimental results are consistent with biased random motion by the bacteria. The purpose of this paper is to discuss a calculation of the predictions of a model previously used to study bacterial chemotaxis in other experimental situations. The details of the experimental setup discussed here are a modification of those used by W & M. This modification significantly reduces the complexity of the calculations necessary to obtain the results while it exhibits the essential aspects of the phenomena being investigated. In addition, the model may be used to make predictions of the results of experiments which have not yet been performed. A set of such experiments is suggested for further research in this area. DESCRIPTION
OF THE MODEL
In earlier work, Lapidus and Schiller [14] developed a model for chemotaxis by bacteria which includes a sensitivity to the concentration of attractant as well as a response to a gradient of attractant concentration. This model has been applied to the study of bands of chemotactic bacteria [15] originally observed by Adler [l]. More recently Holz and Chen [6, 71 have studied the motion of bands of chemotactic bacteria using laser scattering techniques. These latter studies provide significantly more precise experimental data on the band profile as a function of time, which make possible more detailed tests of the mathematical model. The model has also been used to study the motion of populations of bacteria in time-independent concentration gradients [ 141. The mathematical model used here is a development from an earlier model originally suggested by Keller and Segel 191. A number of other authors have used the earlier model to study the formation of traveling bands [ 10, 18, 21-23, 8, 20, 241 and the motion of populations of bacteria in time-independent concentration gradients [25, 19, 1 l- 131. Consider a population of cells with a density b(r, t), where r and t are the position in space and the time at which measurements are made. If the
BACTERIAL CHEMOTAXIS IN FLOWING WATER
81
population density at any point varies in time due to random swimming and chemotactic response, one may write the “equation of continuity” or “equation of conservation of the number of bacteria” as
ab=-, at
J
(1)
‘3
where J is the current density of the bacteria, i.e. the number which pass a unit area per unit time. The current density is composed of two parts: J=J,+J,=-pVb+GbVf,
of bacteria
(2)
where J, is the random (or “diffusion-like”) current and J, is the chemotactic current. The quantities ).Land S are the random and chemotactic motilities, which have units of length*/time (e.g., mm’/min). The quantity f is the sensitivity function, which is a function of the concentration of attractant. On the basis of experiments by Mesibov, Ordal, and Adler [17], Lapidus and Schiller [14] adopted the functional form
where s=s(r, t) is determined by the attracted to amino In the presence adding a term
the concentration of attractant and k is a constant particular organism being studied. (For Escherichia coli acids, k= 10 -3 M.) of flowing water the current J must be modified by
J, =v,b,
(4)
where v, is the (constant) velocity of the water which sweeps along the bacterial population. Thus, the basic equation for the motion of the bacterial population becomes $
=pV”b-6Vb.
Vf-SbV2f-v,,.
Vb.
(5)
If the attractant is not consumed by the bacteria, the attractant will spread by diffusion throughout the chamber. In flowing water the current density for the attractant is given by J= -DVs+v,s, where D is the diffusion
constant
for the attractant
(6) in water. The equation
I. RICHARD
82
LAPIDUS
for the attractant concentration as a function of position and time is then as Z =DV’s-v,.Vs. The diffusion terms in Eqs. (5) and (7), though similar in appearance, have different origins. The attractant, which consists of small molecules, diffuses by random collisions (Brownian motion). On the other hand, the bacteria “diffuse” by swimming in random directions and then changing their direction by “tumbling” [2]. (The effect of actual Brownian motion due to molecular collisions with the bacteria is much smaller.) In order to carry out a numerical calculation which may be compared with the experimental results, the design for an experimental apparatus, which is a modification of that actually used by W & M to obtain experimental data, has been used as the basis for the calculations, Consider a cylindrical chamber through which water flows with a constant velocity of magnitude u,,. The numerical value of v, is fixed for a given experimental run but may be varied by the experimenter for different runs. The influx contains a uniform concentration of bacterial cells. The flow velocity of the effluent is the same as that of the influx, but the density of cells carried out of the chamber is determined by the behavior of the bacteria in the apparatus. At time t -0 a pulse of chemotactic attractant is injected into the chamber. The injection point is taken to be at the center of the chamber, although the exact position does not affect the results except for the numerical details. The pulse then diffuses and is also swept out of the chamber by the flow of water. The experimental apparatus is cylindrically symmetric about an axis through the chamber, and motion perpendicular to the axis averages to zero. Hence, the motion of the bacteria effectively reduces to that in one dimension. Thus, Eqs. (5) and (7) may be rewritten as
as z
a%
=Daxz -oz.
as
In order to obtain solutions to these equations it is necessary to specify the “boundary conditions” at the ends of the chamber and the initial conditions. In the absence of water flow, i.e., for u. =O, the random and chemotactic current of the bacteria and the diffusion current of the attractant leaving the chamber at x= + L are zero. If the inlet and outlet
BACTERIAL
CHEMOTAXIS
IN FLOWING
diameters are small compared tion is still valid, i.e.,
-pE
to the diameter
+6bg
83
WATER
=0
at
of the chamber,
this condi-
x=%L,
(‘0)
and
_&
=o
ax
at
x=?L.
(1’)
But, from Eq. (1 l), the second term in Eq. (10) vanishes. Thus, -pax
ab
=o
at
x=+L.
(12)
The initial conditions at t= 0 are as follows: b(x, 0) = b, throughout the tube; s(x,O)=O except at the point x -0, where a pulse of attractant has been injected. This may be written as s(x,0)=(soL)6(x), where S(X) is the &function with the property J 6(x) dx = 1, and IV= s,, L is the total number of molecules injected into the system. (The cross-sectional area of the chamber, which does not appear, is taken as equal to 1.) Equations (8) and (9) may be solved as follows: Equation (9) is a familiar linear partial differential equation. The solution obtained may be inserted into Eq. (8), which, however, cannot be solved exactly. Since an approximation method is required to solve Eq. (8) the same numerical technique, the Crank-Nicolson method [3], has been used to solve both equations, and a high-speed digital computer has been employed to carry out the cafcuIations. The a-function condition is approximated by setting s=sO at a single grid Iocation at the time t = 0. RESULTS In order to appreciate the significance of the results obtained using our model for the chemotactic behavior of a bacterial population, it is important to distinguish between the different types of parameters used in the model and to articulate the nature of the predictions which are possible. Equations (8) and (9) contain six parameters. The quantities p, 6, and k are properties of the bacteria which are independent of this particular set of experiments. They may be determined by procedures described by Lapidus and Schiller 1141 or from migrating band experiments [6, 71. The quantity D is the diffusion constant for the attractant in water, which is also independent of this set of experiments. Thus, u,, the velocity of the water flow, and so, the concentration of the attractant pulse, are the only variables. But these are determined by the
84
I. RICHARD
LAPIDUS
experimenter and are not adjustable parameters of the theory. The dimensions of the apparatus also enter the calculations. The length L and the cross-sectional area of the chamber are set by the experimenter; they are not variables. In the one-dimensional approximation used here, the crosssectional area does not enter the calculations at all. Thus, if ~1,6, and k are known, there are no adjustable parameters in this theory which are available to fit this set of experiments. Nonetheless, the theory is able to predict the distribution of attractant concentration and bacterial density at all times following the addition of the pulse of attractant. From this complete output it is possible to select a number of useful data for comparison with experimental measurements. For comparison with the data of W & M, it is only necessary to examine the height of the bacterial density peak, as measured in the washout, as a function of time for various values of I+,. In this way one can produce the “data” predicted by the model, which are equivalent to those obtained experimentally, and compare the two. In addition, computer “experiments” beyond those of W & M have been carried out. In particular the height and position of the peak as a function of time have been determined for different values of s,,, while v, is kept constant. It would be of great interest to obtain experimental data which may be compared with these predictions, since they are made in advance of the actual experiments. The numerical results which have been obtained are shown in the figures. For each figure the bacterial parameters have been assigned the values ~=0.5 mm*/min, a-+6 mm*/min, k= 1, and s0 is measured in units of k; D = 0.05 mm* /min. Figure 1 shows a plot of the effluent bacterial density as a function of time for v0 = 10 mm/min. The figure is essentially of the same form as that obtained by W & M. For this figure sa =k. Figure 2 shows the maximum and minimum values of the bacterial effluent as functions of the flow velocity u,,. The relationship follows the same form as that obtained by W & M. For this figure s0 = k. Figure 3 shows the washout time of the maximum and minimum in the bacterial effluent as functions of the flow velocity vO. Again, in agreement with W & M. the washout time decreases with increasing flow velocity. For this figure s0 = k. Figure 4 shows the values of the maximum and minimum in the bacterial effluent as functions of sO. For this figure u0 = 10 mm/mm. Figure 5 shows the washout time of the maximum and minimum in the bacterial effluent as functions of s,,. For this figure u0 = 10 mm/mm. The irregularities in the figures are due to the fact that only a few points in a grid of parameter values were used in the calculations. The maximum and minimum values were selected from this output. The straight segments
BACTERIAL
CHEMOTAXIS
IN FLOWING
85
WATER
I]
%h
6.BB
I!?.@0
10.88
24.88
I 38.68
TIME FIG. 1. Plot of the bacterial density iu the effluent of the apparatus as a function of time for the flow velocity 0, = 10 mm/min. The pulse of attractant is injected at r-0. (&I = w
FIG. 2. Plot of the maximum and minimum functions of the fiow velocity oc. (so =k.)
values
of the bacterial
effluent
as
I. RICHARD
86
?
LAPIDUS
1
wlO.oO
12.08
FI-OW
14.00
RATE
16.00
20.00
le.m
( MM/MIN)
FIG. 3. Plot of the time for washout of the maximum and minimum in the bacterial effluent as functions of the flow velocity oO. (sO = k.)
6 y-2.88
-1.50
LOG
-1.w
S/K)
, -0.58
, 0.W
FIG. 4. Plot of the values of the maximum and minimum in the bacterial effluent as functions of the concentration of the pulse of attractant, s,-,. (oO = 10 mm/min.)
BACTERIAL
FIO. 5.
CHEMOTAXIS
IN FLOWING
WATER
87
Plot of the washout time of the maximum and minimum in the bacterial
effluent functions of the concentration of the
pulse of attractant, s,. (uO- 10 mm/min.)
merely connect the computed points. The exact results must, of course, be smooth curves. Since each point is the result of a calculation which yields a curve similar to Fig. 1, a large amount of computer time would be necessary to smooth out the other figures. DISCUSSION Experiments similar to those of W & M could provide useful tests of the model because the predictions are unambiguous and experimentally varifiable. Since the washout properties of the system are a function of sc, it may be possible to obtain direct measurements of the sensitivity functionf(s) by carrying out a series of experiments for varying values of se. Although the theory, using Eq. (3) has proved successful in predicting the behavior of bacterial populations in fixed gradients [14] and the formation and propagation of traveling bands [ 151, the form of f(s), given by Eq. (3), has not been determined directly. These experiments could provide a means of doing this. Even if the experimentally measured sensitivity function does not agree with the form given in Eq. (3), its measurement would be of great interest because it might then be used to predict bacterial chemotactic behavior in
88
I. RICHARD LAFTDUS
other experimental situations (unless the sensitivity function is very different in different species of bacteria). In order to improve the efficiency of data collection it may be possible to pass the effluent from the chamber directly through a Coulter counter. In this way continuous readings of the density of bacteria in the effluent may be obtained and the data may be recorded immediately. This procedure would appear to be a considerable improvement over the plating method employed by W & M. It should be noted that the experimental procedures of W & M are actually different from those discussed here. The injection procedure used by them probably produced a pulse of attractant which was spread out over a finite volume of space rather than being initially at a point. If the process of spreading of the pulse were due only to diffusion, the time scale would be long compared to the experimental times. In addition, one would expect there to be a considerable amount of mixing of the attractant due to currents perpendicular to the length of the chamber immediately after the injection. The experimental setup used by W h M necessarily confounds two distinct effects. In the analysis presented here only the effective upstream swimming has been considered, while in the W 8z M experiments the difference in flux from the two exit tubes is due to the asymmetry in the setup. Thus, a theoretical analysis of their experiments is considerably more difficult because of the interference between the two aspects of the phenomena. It would be of great interest if the experiments could be repeated using a chamber similar to that described for the calculations presented here. SUMMARY It has been shown that a mathematical model for bacterial chemotaxis which has been previously used to study the response of bacterial populations to fixed attractant gradients and the formation and propagation of bands in capillary tubes, also may be used to explain the behavior of bacteria responding to chemotactic attractants in flowing water. Experimental data of Walsh and Mitchell have been compared with the predictions of the model. In addition, using the mathematical model, new predictions have been made which may be tested experimentally. In particular, it is suggested that the experiments of W & M be repeated using a modified apparatus and for a range of values of the concentration of the pulse of attractant. I wish to thank M. Levandowsky for stimulating discussions of microbial responses to chemical signals. The numerical calculations were carried out at
BACTERIAL
CHEMOTAXIS
IN FLOWING
89
WATER
the Stevens Computer Center. I wish to thank the stuff of the Center for their assistance on numerous occasions. REFERENCES 1 2 3 4 5 6 7
J. Adler, Chemotaxis in bacteria, Science 153:708 (1966). H. C. Berg and D. A. Brown, Chemotaxis in Escherichia dimensional tracking, Nature 239:500 (1972). J. Crank, T?ze Mathemafics of Diffurion, Oxford U.P., 1975.
cdi
analyzed
by three-
F. W. Dahlquist, P. Lovely, and D. E. Koshland, Quantitative analysis of bacterial migration in chemotaxis, Nat. New BioI. 236: 120 (1972). M. A. de Jong, C. van der Drift, and G. D. Vogels, Receptors for chemotaxis in Bacillus subdis, J. BacferioI. 123 :824 (1975). M. Holz and S.-H. Chen, Quasi-elastic light scattering from migrating chemotactic bands of Escherichia coli, Biophys. J. 23: 15 (1978). M. Holz and S.-H. Chen, Spatio-temporal structure of migrating chemotactic bands of Esckrictia coli. I. Traveling band profile, Biophys. J. 26:243 (1979).
8
E. F. Keller and G. M. Odell, Necessary bands, Math. Biosci. 27:309 (1975).
9
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol. 26:399 (1970). E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol. 30:235 (1971).
10 11
and sufficient
conditions
for chemotactic
I. R. Lapidus and R. Schiller, A mathematical model for bacterial Biophys. J. 14:825 (1974). I. R. Lapidus and R. Schiller, Bacterial chemotaxis in a two-dimensional gradient, J. Biol. Phys. 2:205 (1974).
chemotaxis,
13
I. R. Lapidus and R. Schiller, Theorer. Biol. 53:215 (1975).
gradient,
14
I. R. Lapidus and R. Schiller, Model population, Biq0hy.r. J. 16:779 (1976).
15
I. R. Lapidus and R. SchilIer, A model for traveling bands Biophys. J. 22: 1 (1978). A. McVittie and S. A. Zahler, Chemotaxis in myxobacteria,
12
16 17
18 19 20 21 22 23
Bacterial
chemotaxis
in a fixed attractant
for the chemotactic
response
J.
of a bacterial
of chemotactic Nature
attractant
bacteria,
194: 1299 (1%2).
R. Mesibov, G. W. Ordal, and J. Adler, The range of attractant concentrations for bacterial chemotaxis and the threshold and size of response over this range-Weber law and related phenomena, J. Gen. Physiol. 62:203 (1973). R. Nossal, Boundary movement of chemotactic bacterial populations, Moth. Biosci. 13:397 (1972). R. Nossal and G. A. Weiss, Analysis of a densitometry assay in bacterial chemotaxis, J. Theoret. Biol. 41:143 (1973). G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited, J. Theoret. Biol. 56~243 (1976). G. Rosen, On the propagation theory for bands of chemotactic bacteria, Math. Biosci. 20: 185 (1974). G. Rosen, Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria, J. Theoret. Biol. 49:3 11 (1975). G. Rosen and S. Baloga, On the stability of steadily propagating bands of chemotactic bacteria, Mudz. Biosci. 24:273 (1975).
1. RICHARD
90 24
LAPIDUS
27
T. L. Scribner, L. A. Segel, and E. H. Rogers, A numerical study of the formation and propagation of traveling bands of chemotactic bacteria, J. Theoret. Biol. 46: 189. L. A. Segel and J. L. Jackson, Theoretical analysis of chemotactic movements in bacteria, J. Mechanochem. CeN Motili@ 2:25 (1973). F. W. K. Seymour and R. N. Deutsch, Chemotactic responses in motile bacteria, J. Gen. Microbial. 78~287 (1973). S. C. Straley and S. F. Conti, Chemotaxis in Bdellovibrio bacterioww, J. Bacterial.
28
120:549 (1974). F. Walsh and R. Mitchell, Bacterial chemotactic response in flowing water, Microb.
25 26
Ecol. 4: 165 (1978).
in Flowing Water
I. RICHARD LAPIDUS Depariment of Physics, Stew-~ Instiiuie of Technology, Hobokeq
New Jersey 07030
Receiced 27 January 1980
ABSTRACT Chemotactic behavior by bacterial populations is studied using a mathematical model to analyze the response of swimming bacteria to a pulse of chemotactic attractant injected into flowing water. Experiments by Walsh and Mitchell are discussed using this model, and it is shown the experimental results can be understood in terms of the predictions of the model. The observations are consistent with biased random swimming by the bacteria. The model also makes predictions of the results of new experiments which are suggested for future research.
INTRODUCTION Chemotaxis by bacteria has been investigated in a number of different organisms, including Escherichia coli [l, 21, Salmonella typhimurium [4], Bacillus subtilis [5], Bdellovibrio bacteriovorus [27], Myxococcus xanthus [ 161, and others [26]. The experimental techniques have included the Adler [1] assay, the migration of bacterial populations in fixed attractant concentration gradients observed by Dahlquist et al. [4], and the formation and propagation of bands and rings which are generated by the consumption of attractant to maintain a moving chemical gradient [ 11. Recently
MA
TICAL
I. RICHARD LAPIDUS
80
increasing flow rate, and the time to reach this maximum value also decreased approximately linearly. These results may be understood qualitatively by noting that if the flow rate is low, the time for the bacteria to accumulate near the pulse of attractant will be large, and this will give rise to a large peak as the attractant is slowly washed out. On the other hand, if the flow rate is large, the time for the bacteria to accumulate before washout occurs will be short and the peak size will necessarily be small. Despite the fact that it is possible to understand the experimental results obtained by W & M qualitatively, it is of interest to obtain quantitative agreement between a theoretical formulation capable of making numerical predictions and the experimental data. It will be shown that the experimental results are consistent with biased random motion by the bacteria. The purpose of this paper is to discuss a calculation of the predictions of a model previously used to study bacterial chemotaxis in other experimental situations. The details of the experimental setup discussed here are a modification of those used by W & M. This modification significantly reduces the complexity of the calculations necessary to obtain the results while it exhibits the essential aspects of the phenomena being investigated. In addition, the model may be used to make predictions of the results of experiments which have not yet been performed. A set of such experiments is suggested for further research in this area. DESCRIPTION
OF THE MODEL
In earlier work, Lapidus and Schiller [14] developed a model for chemotaxis by bacteria which includes a sensitivity to the concentration of attractant as well as a response to a gradient of attractant concentration. This model has been applied to the study of bands of chemotactic bacteria [15] originally observed by Adler [l]. More recently Holz and Chen [6, 71 have studied the motion of bands of chemotactic bacteria using laser scattering techniques. These latter studies provide significantly more precise experimental data on the band profile as a function of time, which make possible more detailed tests of the mathematical model. The model has also been used to study the motion of populations of bacteria in time-independent concentration gradients [ 141. The mathematical model used here is a development from an earlier model originally suggested by Keller and Segel 191. A number of other authors have used the earlier model to study the formation of traveling bands [ 10, 18, 21-23, 8, 20, 241 and the motion of populations of bacteria in time-independent concentration gradients [25, 19, 1 l- 131. Consider a population of cells with a density b(r, t), where r and t are the position in space and the time at which measurements are made. If the
BACTERIAL CHEMOTAXIS IN FLOWING WATER
81
population density at any point varies in time due to random swimming and chemotactic response, one may write the “equation of continuity” or “equation of conservation of the number of bacteria” as
ab=-, at
J
(1)
‘3
where J is the current density of the bacteria, i.e. the number which pass a unit area per unit time. The current density is composed of two parts: J=J,+J,=-pVb+GbVf,
of bacteria
(2)
where J, is the random (or “diffusion-like”) current and J, is the chemotactic current. The quantities ).Land S are the random and chemotactic motilities, which have units of length*/time (e.g., mm’/min). The quantity f is the sensitivity function, which is a function of the concentration of attractant. On the basis of experiments by Mesibov, Ordal, and Adler [17], Lapidus and Schiller [14] adopted the functional form
where s=s(r, t) is determined by the attracted to amino In the presence adding a term
the concentration of attractant and k is a constant particular organism being studied. (For Escherichia coli acids, k= 10 -3 M.) of flowing water the current J must be modified by
J, =v,b,
(4)
where v, is the (constant) velocity of the water which sweeps along the bacterial population. Thus, the basic equation for the motion of the bacterial population becomes $
=pV”b-6Vb.
Vf-SbV2f-v,,.
Vb.
(5)
If the attractant is not consumed by the bacteria, the attractant will spread by diffusion throughout the chamber. In flowing water the current density for the attractant is given by J= -DVs+v,s, where D is the diffusion
constant
for the attractant
(6) in water. The equation
I. RICHARD
82
LAPIDUS
for the attractant concentration as a function of position and time is then as Z =DV’s-v,.Vs. The diffusion terms in Eqs. (5) and (7), though similar in appearance, have different origins. The attractant, which consists of small molecules, diffuses by random collisions (Brownian motion). On the other hand, the bacteria “diffuse” by swimming in random directions and then changing their direction by “tumbling” [2]. (The effect of actual Brownian motion due to molecular collisions with the bacteria is much smaller.) In order to carry out a numerical calculation which may be compared with the experimental results, the design for an experimental apparatus, which is a modification of that actually used by W & M to obtain experimental data, has been used as the basis for the calculations, Consider a cylindrical chamber through which water flows with a constant velocity of magnitude u,,. The numerical value of v, is fixed for a given experimental run but may be varied by the experimenter for different runs. The influx contains a uniform concentration of bacterial cells. The flow velocity of the effluent is the same as that of the influx, but the density of cells carried out of the chamber is determined by the behavior of the bacteria in the apparatus. At time t -0 a pulse of chemotactic attractant is injected into the chamber. The injection point is taken to be at the center of the chamber, although the exact position does not affect the results except for the numerical details. The pulse then diffuses and is also swept out of the chamber by the flow of water. The experimental apparatus is cylindrically symmetric about an axis through the chamber, and motion perpendicular to the axis averages to zero. Hence, the motion of the bacteria effectively reduces to that in one dimension. Thus, Eqs. (5) and (7) may be rewritten as
as z
a%
=Daxz -oz.
as
In order to obtain solutions to these equations it is necessary to specify the “boundary conditions” at the ends of the chamber and the initial conditions. In the absence of water flow, i.e., for u. =O, the random and chemotactic current of the bacteria and the diffusion current of the attractant leaving the chamber at x= + L are zero. If the inlet and outlet
BACTERIAL
CHEMOTAXIS
IN FLOWING
diameters are small compared tion is still valid, i.e.,
-pE
to the diameter
+6bg
83
WATER
=0
at
of the chamber,
this condi-
x=%L,
(‘0)
and
_&
=o
ax
at
x=?L.
(1’)
But, from Eq. (1 l), the second term in Eq. (10) vanishes. Thus, -pax
ab
=o
at
x=+L.
(12)
The initial conditions at t= 0 are as follows: b(x, 0) = b, throughout the tube; s(x,O)=O except at the point x -0, where a pulse of attractant has been injected. This may be written as s(x,0)=(soL)6(x), where S(X) is the &function with the property J 6(x) dx = 1, and IV= s,, L is the total number of molecules injected into the system. (The cross-sectional area of the chamber, which does not appear, is taken as equal to 1.) Equations (8) and (9) may be solved as follows: Equation (9) is a familiar linear partial differential equation. The solution obtained may be inserted into Eq. (8), which, however, cannot be solved exactly. Since an approximation method is required to solve Eq. (8) the same numerical technique, the Crank-Nicolson method [3], has been used to solve both equations, and a high-speed digital computer has been employed to carry out the cafcuIations. The a-function condition is approximated by setting s=sO at a single grid Iocation at the time t = 0. RESULTS In order to appreciate the significance of the results obtained using our model for the chemotactic behavior of a bacterial population, it is important to distinguish between the different types of parameters used in the model and to articulate the nature of the predictions which are possible. Equations (8) and (9) contain six parameters. The quantities p, 6, and k are properties of the bacteria which are independent of this particular set of experiments. They may be determined by procedures described by Lapidus and Schiller 1141 or from migrating band experiments [6, 71. The quantity D is the diffusion constant for the attractant in water, which is also independent of this set of experiments. Thus, u,, the velocity of the water flow, and so, the concentration of the attractant pulse, are the only variables. But these are determined by the
84
I. RICHARD
LAPIDUS
experimenter and are not adjustable parameters of the theory. The dimensions of the apparatus also enter the calculations. The length L and the cross-sectional area of the chamber are set by the experimenter; they are not variables. In the one-dimensional approximation used here, the crosssectional area does not enter the calculations at all. Thus, if ~1,6, and k are known, there are no adjustable parameters in this theory which are available to fit this set of experiments. Nonetheless, the theory is able to predict the distribution of attractant concentration and bacterial density at all times following the addition of the pulse of attractant. From this complete output it is possible to select a number of useful data for comparison with experimental measurements. For comparison with the data of W & M, it is only necessary to examine the height of the bacterial density peak, as measured in the washout, as a function of time for various values of I+,. In this way one can produce the “data” predicted by the model, which are equivalent to those obtained experimentally, and compare the two. In addition, computer “experiments” beyond those of W & M have been carried out. In particular the height and position of the peak as a function of time have been determined for different values of s,,, while v, is kept constant. It would be of great interest to obtain experimental data which may be compared with these predictions, since they are made in advance of the actual experiments. The numerical results which have been obtained are shown in the figures. For each figure the bacterial parameters have been assigned the values ~=0.5 mm*/min, a-+6 mm*/min, k= 1, and s0 is measured in units of k; D = 0.05 mm* /min. Figure 1 shows a plot of the effluent bacterial density as a function of time for v0 = 10 mm/min. The figure is essentially of the same form as that obtained by W & M. For this figure sa =k. Figure 2 shows the maximum and minimum values of the bacterial effluent as functions of the flow velocity u,,. The relationship follows the same form as that obtained by W & M. For this figure s0 = k. Figure 3 shows the washout time of the maximum and minimum in the bacterial effluent as functions of the flow velocity vO. Again, in agreement with W & M. the washout time decreases with increasing flow velocity. For this figure s0 = k. Figure 4 shows the values of the maximum and minimum in the bacterial effluent as functions of sO. For this figure u0 = 10 mm/mm. Figure 5 shows the washout time of the maximum and minimum in the bacterial effluent as functions of s,,. For this figure u0 = 10 mm/mm. The irregularities in the figures are due to the fact that only a few points in a grid of parameter values were used in the calculations. The maximum and minimum values were selected from this output. The straight segments
BACTERIAL
CHEMOTAXIS
IN FLOWING
85
WATER
I]
%h
6.BB
I!?.@0
10.88
24.88
I 38.68
TIME FIG. 1. Plot of the bacterial density iu the effluent of the apparatus as a function of time for the flow velocity 0, = 10 mm/min. The pulse of attractant is injected at r-0. (&I = w
FIG. 2. Plot of the maximum and minimum functions of the fiow velocity oc. (so =k.)
values
of the bacterial
effluent
as
I. RICHARD
86
?
LAPIDUS
1
wlO.oO
12.08
FI-OW
14.00
RATE
16.00
20.00
le.m
( MM/MIN)
FIG. 3. Plot of the time for washout of the maximum and minimum in the bacterial effluent as functions of the flow velocity oO. (sO = k.)
6 y-2.88
-1.50
LOG
-1.w
S/K)
, -0.58
, 0.W
FIG. 4. Plot of the values of the maximum and minimum in the bacterial effluent as functions of the concentration of the pulse of attractant, s,-,. (oO = 10 mm/min.)
BACTERIAL
FIO. 5.
CHEMOTAXIS
IN FLOWING
WATER
87
Plot of the washout time of the maximum and minimum in the bacterial
effluent functions of the concentration of the
pulse of attractant, s,. (uO- 10 mm/min.)
merely connect the computed points. The exact results must, of course, be smooth curves. Since each point is the result of a calculation which yields a curve similar to Fig. 1, a large amount of computer time would be necessary to smooth out the other figures. DISCUSSION Experiments similar to those of W & M could provide useful tests of the model because the predictions are unambiguous and experimentally varifiable. Since the washout properties of the system are a function of sc, it may be possible to obtain direct measurements of the sensitivity functionf(s) by carrying out a series of experiments for varying values of se. Although the theory, using Eq. (3) has proved successful in predicting the behavior of bacterial populations in fixed gradients [14] and the formation and propagation of traveling bands [ 151, the form of f(s), given by Eq. (3), has not been determined directly. These experiments could provide a means of doing this. Even if the experimentally measured sensitivity function does not agree with the form given in Eq. (3), its measurement would be of great interest because it might then be used to predict bacterial chemotactic behavior in
88
I. RICHARD LAFTDUS
other experimental situations (unless the sensitivity function is very different in different species of bacteria). In order to improve the efficiency of data collection it may be possible to pass the effluent from the chamber directly through a Coulter counter. In this way continuous readings of the density of bacteria in the effluent may be obtained and the data may be recorded immediately. This procedure would appear to be a considerable improvement over the plating method employed by W & M. It should be noted that the experimental procedures of W & M are actually different from those discussed here. The injection procedure used by them probably produced a pulse of attractant which was spread out over a finite volume of space rather than being initially at a point. If the process of spreading of the pulse were due only to diffusion, the time scale would be long compared to the experimental times. In addition, one would expect there to be a considerable amount of mixing of the attractant due to currents perpendicular to the length of the chamber immediately after the injection. The experimental setup used by W h M necessarily confounds two distinct effects. In the analysis presented here only the effective upstream swimming has been considered, while in the W 8z M experiments the difference in flux from the two exit tubes is due to the asymmetry in the setup. Thus, a theoretical analysis of their experiments is considerably more difficult because of the interference between the two aspects of the phenomena. It would be of great interest if the experiments could be repeated using a chamber similar to that described for the calculations presented here. SUMMARY It has been shown that a mathematical model for bacterial chemotaxis which has been previously used to study the response of bacterial populations to fixed attractant gradients and the formation and propagation of bands in capillary tubes, also may be used to explain the behavior of bacteria responding to chemotactic attractants in flowing water. Experimental data of Walsh and Mitchell have been compared with the predictions of the model. In addition, using the mathematical model, new predictions have been made which may be tested experimentally. In particular, it is suggested that the experiments of W & M be repeated using a modified apparatus and for a range of values of the concentration of the pulse of attractant. I wish to thank M. Levandowsky for stimulating discussions of microbial responses to chemical signals. The numerical calculations were carried out at
BACTERIAL
CHEMOTAXIS
IN FLOWING
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WATER
the Stevens Computer Center. I wish to thank the stuff of the Center for their assistance on numerous occasions. REFERENCES 1 2 3 4 5 6 7
J. Adler, Chemotaxis in bacteria, Science 153:708 (1966). H. C. Berg and D. A. Brown, Chemotaxis in Escherichia dimensional tracking, Nature 239:500 (1972). J. Crank, T?ze Mathemafics of Diffurion, Oxford U.P., 1975.
cdi
analyzed
by three-
F. W. Dahlquist, P. Lovely, and D. E. Koshland, Quantitative analysis of bacterial migration in chemotaxis, Nat. New BioI. 236: 120 (1972). M. A. de Jong, C. van der Drift, and G. D. Vogels, Receptors for chemotaxis in Bacillus subdis, J. BacferioI. 123 :824 (1975). M. Holz and S.-H. Chen, Quasi-elastic light scattering from migrating chemotactic bands of Escherichia coli, Biophys. J. 23: 15 (1978). M. Holz and S.-H. Chen, Spatio-temporal structure of migrating chemotactic bands of Esckrictia coli. I. Traveling band profile, Biophys. J. 26:243 (1979).
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E. F. Keller and G. M. Odell, Necessary bands, Math. Biosci. 27:309 (1975).
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E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol. 26:399 (1970). E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol. 30:235 (1971).
10 11
and sufficient
conditions
for chemotactic
I. R. Lapidus and R. Schiller, A mathematical model for bacterial Biophys. J. 14:825 (1974). I. R. Lapidus and R. Schiller, Bacterial chemotaxis in a two-dimensional gradient, J. Biol. Phys. 2:205 (1974).
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I. R. Lapidus and R. SchilIer, A model for traveling bands Biophys. J. 22: 1 (1978). A. McVittie and S. A. Zahler, Chemotaxis in myxobacteria,
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R. Mesibov, G. W. Ordal, and J. Adler, The range of attractant concentrations for bacterial chemotaxis and the threshold and size of response over this range-Weber law and related phenomena, J. Gen. Physiol. 62:203 (1973). R. Nossal, Boundary movement of chemotactic bacterial populations, Moth. Biosci. 13:397 (1972). R. Nossal and G. A. Weiss, Analysis of a densitometry assay in bacterial chemotaxis, J. Theoret. Biol. 41:143 (1973). G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited, J. Theoret. Biol. 56~243 (1976). G. Rosen, On the propagation theory for bands of chemotactic bacteria, Math. Biosci. 20: 185 (1974). G. Rosen, Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria, J. Theoret. Biol. 49:3 11 (1975). G. Rosen and S. Baloga, On the stability of steadily propagating bands of chemotactic bacteria, Mudz. Biosci. 24:273 (1975).
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LAPIDUS
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T. L. Scribner, L. A. Segel, and E. H. Rogers, A numerical study of the formation and propagation of traveling bands of chemotactic bacteria, J. Theoret. Biol. 46: 189. L. A. Segel and J. L. Jackson, Theoretical analysis of chemotactic movements in bacteria, J. Mechanochem. CeN Motili@ 2:25 (1973). F. W. K. Seymour and R. N. Deutsch, Chemotactic responses in motile bacteria, J. Gen. Microbial. 78~287 (1973). S. C. Straley and S. F. Conti, Chemotaxis in Bdellovibrio bacterioww, J. Bacterial.
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120:549 (1974). F. Walsh and R. Mitchell, Bacterial chemotactic response in flowing water, Microb.
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