J. theor. Biol. (1975) 53, 215-222

Bacterial Chemotaxis in a Fixed Attractant Gradient I. RICHARD LAPIDUS AND RALPH SCHILLER Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, U.S.A. (Received 12 August 1974, and in revised form 13 November 1974)

A differential equation describing the motion of a population of chemotactic bacteria in an exponential gradient of attractant has been solved analytically. Specific predictions are made of the behavior of the bacterial distribution as a function of space and time. The time behavior of several experimental parameters has also been calculated. Experimental tests of these predictions will be important in assessing the validity of this mathematical description of bacterial chemotaxis. 1. Introduction Recently Dahlquist, Lovely & Koshland (1972) (DLK) carried out a series of experiments to study the migration of Salmonella typhimurium in timeindependent concentration of L-serine. The motion of the bacterial population takes place in a capillary tube and is essentially one-dimensional. Keller & Segel (1971) proposed a mathematical model to describe the macroscopic space-time behavior of a population of chemotactic bacteria. In this model, bacterial flow is described by two currents. One represents random diffusive motion and the second the coherent chemotactic movement produced by chemical gradients. Their bacterial flow equation was applied to one of the DLK experiments by Segel & Jackson (1973) and by Nossal & Weiss (1973). These authors showed that for this particular experiment the solution of the Keller-Segel equation confirmed two important observations of DLK; the peak value of the bacterial density profile grows as the square root of the time and the area under the bacterial density distribution peak increases linearly with the time. In a recent paper Lapidus & Schiller (1974) (LS) pointed out that the DLK experiment, since it ran for only 40 min, did not provide an adequate test of the Keller-Segel theory. While the experiment may have proved the significance of random motion for chemotactic flow, it provided little information regarding the nature of the chemotactic current. This is because ordinary diffusion theory with a known flow of bacteria into the origin can 215

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reproduce the two cited observations of the DLK experiment. The detailed form of the chemotactic current is almost irrelevant for the first 40 min of the DLK experiment because its role in the flow may be characterized by a single constant parameter, the rate at which bacteria move into the central region. Had the DLK experiment run for a longer time, the precise form of the chemotactic current would have become increasingly important. Unfortunately, this latter period of the system’s approach towards its stationary state was neither examined experimentally nor could it be analyzed in the theoretical treatments of Segel & Jackson (1973) or Nossal & Weiss (1973). LS showed that by a more appropriate choice of boundary conditions, it is possible to find solutions for all time of the Keller-Segel equation as applied to the DLK experiment. These calculations confirmed the predictions made in the earlier papers of Segal & Jackson (1973) and Nossal & Weiss (1973), but only for the 40 min to which DLK limited their experiment. For later times a different time dependence for the peak and the area was obtained. In addition, with these solutions it was possible to trace the evolution in time of several other parameters amenable to experimental measurement. In our view, at this stage of our knowledge, a number of possible chemotactic forces could be responsible for the general flow and the two predicted observations. We believe that the weight of the experimental evidence tends to support the broad features of the Keller-Segel description of bacterial chemotaxis. However, in the absence of more detailed experimental proof, this must be a preliminary judgment. Should the additional observations suggested in this paper confirm the Keller-Segel theory, then it would be important to answer the question: “What biochemical and biophysical mechanisms in individual bacteria would lead to the apparent ensemble response?” On the other hand, if the theoretical predictions are not confirmed, then one must ask, “Is the failure of the theory to agree with the experimental observations due to the form of the assumed force, or is the general approach of the model inadequate?” Additional experimental evidence should help make it possible to pose the appropriate question. In this paper we propose another experimental test of the Keller-Segel equation which to our knowledge has not been performed as yet. The conditions are similar to the previously mentioned DLK experiment, except that in this experiment the attractant concentration has an exponential gradient throughout the length of the capillary tube. For this problem we have found an exact analytical solution of the Keller-Segel equation in the form of a rapidly converging infinite series. We have plotted the time dependence of several different measurable parameters which. characterize the space-time behavior of the bacterial density. These are related to the variables discussed by Lapidus & Schiller (1974). The forms of corresponding curves

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for both experiments are similar, except that in our proposed experiment, the time scale for the evolution of the system towards its stationary st.ate is shortened considerably. We believe that if these new experiments were carried out the nature of the bacterial migration in chemotaxis would be better understood. 2. Chemotactic Equations The Keller-Segel equations describing a chemotactic bacterial population in a one-dimensional attractant gradient have been discussed previously (Keller & Segel, 1971; Lapidus & Schiller, 1974; Nossal & Weiss, 1973; Segel & Jackson, 1973). The quantity b(x, t) is the density of bacteria and J(x, t) is the current density satisfying the “continuity equation”

with

J=-,g+66-

a In (s) ax .

(lb)

We consider an attractant distribution given by s(x) = so exp (- ax), t2) for 0 < x 5 L, where se and a are experimental constants and L is the length of the capillary tube. In this case, equations (1) become db a2b -=LCp-V>;’ at

o’b

(3)

where p is the bacterial “motility” and v = a6 is the “chemotactic velocity”. The boundary conditions for equation (3) are

at x = 0 and x = L. At t = 0 b(x, 0) = b, = constant.

(5) Equations (3) and (4) may be solved by the standard method of “separation of variables”. The complete solution of equations (3), (4) and (5) is b(x, t) = 6, exp( -2x/1)+

5 c, exp (- t/r[l +(n~n/L)~]) tl=l x exp (-x/lz>[sin (mx/L)-(nni/L)

x cos (rim/L)],

(6)

where i = 2p/v, z = 4p/c2, c,, = (4h,L/;1)(1-(-

1)” exp (L/A).(n7rA/L)/[l

+(~mn/L)~]“)

(7)

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and b, = (2b,L/1)/[1-exp

For long times, the time-dependent

(-2L/,Q].

(8)

terms vanish and the solution is

b(x, co) = b, exp (-2x/Q.

(9)

It is of interest to consider some of the properties of the time-independent solution. For small values of v b I 2 bo(lfU4,

(10)

while for large values of z, b 1 z 2b,L/1.

The “width” of the peak at the origin is obtained by setting b(x,) and solving for x = x,,. We obtain

x0 = WMn Gd-ln D-exp (-s>l>,

(11) = b, (12)

where g = 2LliL = Lvlp. For small values of v, x0 = L/2; for large values of U, x0 -+ L In (q)/q -+ 0. Plots of equations (8) and (12) are given in Fig. 4. The total number of bacteria which have migrated into the steady-state peak is given by XD

N =.f [b(x)-b,] 0

dx =r[b,

exp(-2x/2)-b,]

dx

0

= b,L/y){q/(l-exp(-q))-l-ln(~)+ln[l--exp(-y)]}. (13) This reduces to N = 0 for small values of v and N = b,L for large values of v. A plot of equation (13) is given in Fig. 5.

3. Results

The infinite series solution given by equation (6) converges rapidly. Thus, for reasonable experimental times, only the first few terms in the expansion contribute significantly to b(x, t). A number of deductions may be drawn from the solution and we have plotted these in Figs l-5. The numerical calculations have been carried out using the experimental parameters taken from DLK: p = O-333 mm’/min, v = 0.168 tnm/min and L = 20 mm. Figure 1 is a plot of the bacterial density profile at various times: t = 0, t = 10 min, t = 30 min, and the time-independent solution (t -+ co). We note that the peak at the origin increases until the time-independent exponential distribution is reached.

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-5 bo

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.

0

0.5

0

I.0

X/L

FIG. 1. Plot of the bacterial density profile at different values of time: t = 0, t = 10 min,

t = 30 min, t + co.

10

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‘.O

t(min)

FIG. 2. Plots of the density at the peak, b(0, t)/&, the-density at the end of the tube, b(L, t)/b,, and the width of the peak, x,/L, as functions of time.

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40

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tImin)

FIG. 3. Plot of the area under the peak as a function of time. The area is a measure of the number of bacteria which have migrated into the peak.

0.5

0.4

0.3 x0 i0.2

0.1 c

0

0

FIG. 4. Plot of the time-independent functions of q = X/l, = Lvju.

height, bl/bo.

and width, solL, of the peak as

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N b,L

FIG. 5. Plot of the total number of bacteria which ultimately migrate into the peak as a function of rj = 2Ljl = Lvjp.

The density at the peak, x = 0, and at the end of the tube, x = L, are plotted in Fig. 2 as functions of time. These quantities are directly measurable and the experimental results can be compared with the theory. For short times the height of the peak grows as the square root of the time. The “width” of the peak, x,,, is also shown in Fig. 2. This quantity is defined as the value of x at which the bacterial density is equal to the constant bacterial density at t = 0; i.e. b(x,) = bO. In Fig. 3 the area under the peak is plotted as a function of time. The area is proportional to the number of bacteria which have migrated into the peak. This quantity also is obtained experimentally. For short times the number grows linearly with the time. After a long time the bacterial density approaches a stationary distribution, where the specific form of the distribution is determined by the nature of the chemotactic force. In Fig. 4 we have plotted the time independent height and width of the steady-state peak region as a function of q = 2L/A= h/p. The fraction of bacteria which ultimately migrate into the peak is plotted in Fig. 5. The final distribution and its dependence on q are additional checks on the validity of the Keller-Segel theory. Finally, we note that the integral of each of the functions&(x) vanishes. Thus, the total number of bacteria in the capillary tube is determined only

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by the time-independent solution. The time-dependent terms reflect the movement, or redistribution, of the bacterial population due to the chemotactic force and diffusion. Since the magnitude of these terms decreases rapidly with increasing values of n, fewer terms are needed as t increases. When t --) co all the terms in the time-dependent series vanish and the timeindependent solution describes the ultimate configuration of the bacterial density. 4. Conclusions We have presented the results of a calculation describing the motion of a population of chemotactic bacteria in a fixed exponential nutrient gradient. The theory makes a number of predictions which may be compared with experiment. If the theoretical predictions and experimental results conflict, then it will be necessary to re-examine the source of the limited agreement previously discussed in the literature. On the other hand, if the theory is confirmed by the experiments it will be necessary to seek an explanation of its success in terms of the fundamental mechanisms of bacterial chemotaxis including receptor activity, intracellular transport, intracellular communication and genetics. The numerical calculations were carried out at the Stevens Computer Center. REFERENCES DAHLQUIST, F. W., LOVELY, P. & KOSHLAND, D. E., JR. (1972). Nature New Biol. KELLER, E. F. & SEGEL, L. A. (1971). J. theor. Biol. 30,235. LAPIDUS. I. R. & SCHILLER. R. (1974). BioDhvs. J. 14, 825. NOSSAL,‘R. & WEISS, G. A: (19?3). J: theoi. &ok 41,‘143. SEGEL, L. A. & JACKSON, J. L. (1973). J. Mechdnochem. Cell Motility 2,25.

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Bacterial chemotaxis in a fixed attractant gradient.

J. theor. Biol. (1975) 53, 215-222 Bacterial Chemotaxis in a Fixed Attractant Gradient I. RICHARD LAPIDUS AND RALPH SCHILLER Department of Physics, S...
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