J. theor. Biol. (1975) 50,477-496
Statistical Measures of Bacterial Motility
and Chemotaxis
PJITER s. LOVELY Department of Physics, and Lwtitute of Molecular Biology, University of Oregon, Eugene, Oregon 97403, U.S.A.
F. W. DAHLQUIST Department of Chemistry, and Institute qf Molecular Biology, University of Oregon, Eugene, Oregon 97403, U.S.A. (Received 23 April 1974, and in revisedfirm
12 August 1974)
This paper presents a model for the three-dimensional microscopic behavior of motile bacteria, and relates its phrameters to five practical measures of bacterial motility and chemotti (direction correlation function, diffusion constant, persistence time, average velocity, and up/down ratio). The attractant gradient depen@encesof persistence time, average velocity, and up/down ratio are reMed to a single function describing the gradient sensing mechanism. 1. Introduction
Much interest in recent literature has been devoted to physical measurements of the behavior of motile bacteria (Adler & D&l, 1967; Dahlquist, Lovely & Koshland, 1972; Berg & Brown, 1972; Macnab & Koshland, 1973), with the aim of understanding the molecular mechanisms of motihty,.especially during chemotaxis. An experiment is more interesting if it yields detailed microscopic information, such as can be found using the tracking techniques of Berg (1971). But experiments done with less detailed tracking (Lovely, Dahlquist, Macnab & Koshland, 1974) or with macroscopic measurement of concentration of bacteria at different positions in a solution (Dahlquist et al., 1972), can still yield valuable microscopic information, provided the results of these experiments can be related mathematically to a detailed microscopic description of bacterial behavior. It appears worthwhile to relate certain measurable statistical quantities to the parameters of a model for bacterial motion suggested by the tracking data of Berg P Brown (1972) for Escheriehika coli. This model, described below, is not very restrictive, in that it leaves the probability distribution for turning angles and the direction dependence of 477 31 T.B.
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the mean trajectory duration unspecified. It is merely a logical idealization of the experimental evidence, which simplifies analysis, without, one hopes, suppressing any of the crucial features. 2. The Model Let us approximate the three-dimensional motion of a bacterium by making the following assumptions : (a) The path of a bacterium is a sequence of straight line trajectories joined by instantaneous turns, each trajectory being characterized by a speed, direction, and time duration. (b) All trajectories have the same constant speed. (c) When a bacterium turns, its choice of a new direction is governed by a probability distribution which is azimuthally symmetric about the initial direction. (d) The angle between two successive trajectories is governed by a probability distribution which is independent of any other information, such as the direction or duration of the initial trajectory. (e) The duration of a trajectory is governed by a probability distribution which may depend on the direction of that trajectory, but on no other information, such as the parameters of previous trajectories. (f) For any particular direction, the probability density function for the duration of a trajectory is a decaying exponential. Assumptions (a) and (b) are not perfect, and their degree of validity may be judged from the data of Berg & Brown (1972). In most of what follows, trajectories and turns correspond respectively to “runs” and “twiddles”, as defined quantitatively by Berg & Brown (1972). However, as is shown below, rotational diffusion during runs can be accommodated by this model when there is no chemotaxis, by letting a trajectory be much shorter than a run. Also, the non-zero duration of a twiddle can be included in the calculation of the diffusion constant. The data of Berg & Brown (1972) are consistent with assumption (c), and support (d), (e) and (f). They include measurements of a distribution for the angle between successive trajectories, and of a nearly exponential distribution of trajectory durations. We will not need assumption (f) except for discussing the direction correlation function. This assumption is equivalent to saying that the probability of turning in any small time interval depends only on the direction of travel at that time, and not on what happens at any other time. For the isotropic case it means that the number of turns in any time interval is governed by a Poisson distribution. For discussing persistence time, average
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velocity and up/down ratio, we need only to assume that the first moment (mean duration) of the duration probability function exists. For diffusion we must require that the second moment (mean square duration) exists also. Chemotaxis arises from the dependence of’ mean trajectory duration on direction in assumption (e). While the data shaw that this dependence exists, they are insufficient to describe it in detail. It is shown below how persistence time, average velocity and up/down ratio can give information on this dependence. Statistical measures which depend on large numbers of bacteria are complicated by the fact that all the bacteria are not identical. Even though each bacterium may obey this model, they will hot all have the same speed or other parameters. To describe the behavior of an ensemble of independent bacteria with different properties is straightforward and will not be discussed here; but the problem must be borne in mind when interpreting experiments. In terms of this model, five different statistical measures, each of which has been studied experimentally, are described in detail below. The results are summarized in the Discussion and in Table 1. TABLE 1 Summary of formulae relating statistical measures to microscopic behavior information
t+ =$T(u)du=;;jF(x)dx
R = j UT(U) du j xF(x) dx = -1 - 7. J UT(U) du I0 xF(x) dx 0
Correlationtime T.. diffusionconstantD, persistence time t+, averagevelocity VA and up/downratio R are relatedto the meanisotropicqajectory duration T, the direction dependent meandurationT(u), the speedO,the meanturn angle4, andthe meancosinea. The dependence of t+, VA and R on the proportional&tractant gradienty isexpressed in termsof the function F describingthe dependence of meantrajectory duration on the attmctant concentrationand its time derivative.(The concentrationdependence of F is not shown.) 310
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3. Direction Correlation Function We characterize the instantaneous direction of motion of a bacterium by a time-dependent unit vector a(z). We define the direction correlation function by CO> = W)-W>, where < ) means the expectation, averaging over all initial directions a(O), and all possible subsequent paths. Although C(t) can have meaning during chemotaxis, we consider here only cases where all directions are equivalent. Clearly we do not need then to average over initial directions. We must average over the number of trajectories between zero and t; over the angles between these trajectories; and over the azimuthal angles. We do these in reverse order. Suppose n turns occur between 0 and t, and that the n+ 1 trajectories they link have directions a,, al, . . ., a,. Let a,*a, = CI,, a, *a, = CQ, . . ., an-l*an = ~1,. Then, denoting azimuthal angle averaging by ( )+, we have (a(O)*a(t))+, = (a0*a,)4
= alclz . . . a,.
Here we have used an argument familiar in polymer chemistry (Flory, 1969). Averaging now over angles between trajectories (denoted by a subscript $ on the brackets) we have G@)MO>,,
Q = a”,
(1)
where a is the mean cosine of the angle between successive trajectories. This follows because the angles are independent. Finally, averaging over the number of turns which occur between 0 and t, we have C(t) = P,a”+P,a’+P,a2+.
. . +P,a”+.
. .,
where P, is the probability that n turns occur in time t. This sum does not have a simple analytic form in general; but for the specific case of Poisson statistics, Pn has the form p _ (tP)” n n!
e-t/T ’
and the sum is easy to carry out, since ~ ifi (t’T)“a” n=o n!
= ew
This yields C(t) = esf”c,
(2)
MOTILITY
where the correlation
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time is T zc = (1-a)
(3)
Here T is the mean duration of a trajectory. The formula (2) applies to positive t. To include negative, t should be replaced by It 1. It can also be derived simply by starting with the Poisson assumption and stating that C(t + At) = C(t)C(At), which yields a differential equation. We use the above method because ‘it illustrates the complicated nature of the problem in the absence of Poisson statistics. Equation (2) can accommodate the rotational diffusion during runs observed by Berg & Brown (1972). Instead of considering a run to be a single straight line trajectory, we can describe it as a sequence of many very short trajectories. The Poisson process described by T and a may be considered as two simultaneous independent Poisson processes, one of largeangle turns (twiddles) with mean cosine a, at a slow rate l/T,, and another process of small-angle turns (rotational diffusion) with mean cosine cr, at a rapid rate l/T,. Using (3), it is easy to deduce that for this combined process 1 -=A,1 (4) 7, *t 2,’ where 2 T, 5 = G-cr,’ Tt =i--at The “relaxation rates” add in equation (4), as we might expect intuitively for two independent processes. In practical cases, T, and 1 -a, might both be itinitesimal, so that r, is the only meaningful parameter for rotational diffusion. It is worth noting here that we would be unable to describe rotational diffusion during chemotaxis in terms of the model presented in this paper. If one turning rate is isotropic and the other is not, assumption (d) is violated. The direction correlation function can be measured by tracking bacteria with less precision than would be required to record a(t) in detail, because if the assumptions (a)-(f) are obeyed, it is related to another function, the displacement direction correlation function, defined below. Suppose we cannot measure the instantaneous position of a bacterium with enough resolution in time and space to irecord a(r), but that we can record a rough position at regular time intervals, where the interval r between measurements is perhaps comparable to the correlation time. We assume our position measurements are good enough to give a reasonably accurate measurement of the vector displacement between two position measurements.
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We define the displacement direction b(t,, tz) to be a unit vector pointing from the position at time t, to the position at time tz. And we define the displacement direction correlation function to be C,(t) = (b(0, z).b(t, t+z)>. We will discuss only the case where t > z, so that the time intervals (0, r) and (t, t+ T) are disjoint. Azimuthal angle symmetry then insures that components of h(0, r) perpendicular to the final direction a(r) average to zero, as do components of b(t, t+ z) perpendicular to the initial direction a(t). Saving only the parallel components, we can write
C,tO= (MO, 3 *at+(~)] -[atOW>-btt, t + $I> = (WA ~).at~))(a(~).atr))(a(~).b(r,t+4>,
(5) since the three quantities are uncorrelated. The factor in the middle is
N */3> 1+(xo2/
- l)a
1-rx ’ where u now denotes the mean projection of one bond direction on the next. This hypothetical polymer example superimposes on our model for bacterial motion. Replacing n by t/T where T is the mean duration of a trajectory, and interpreting a as in (l), (11) combined with (12) yields D _ 02> 1 +w>2/(~2~ - 1P , l-a 6T where (2’) is the mean square distance traveled between turns, and (I) is the mean distance. This equation can also be #found using the random walk treatment of Patlak (1953). If each new direction is chosen at random, tc = 0 and this reduces to a familiar expression for D in terms of mean square free path and flight time (Chandrasekhar, 1943). For the case where the speed is constant, the formula reduces to D U’S 1+(2T”/S+ 1)cr =(13) 6T l-a ’ where S is the mean square duration of a trajectory, and u is the speed. If assumption (f) holds, it is easy to show that S = 2T2, so the diffusion constant takes the form v2T D (14) = giq*
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Note that by equation (3) we have
D = +=zc,
(15)
which holds even when rotational diffusion is included, as described before. The finite duration of a twiddle can be accommodated by assuming that zero distance is traversed during that time, and replacing n in (12) by t/(T+T,) rather than t/T, where Tl is the mean twiddle duration. This multiplies the expression (14) by T/(T+ Tl). 5. Persistence Time
Macnab & Koshland (1973) have defined persistence time and persistence number, and made measurements of persistence time, in the presence and absence of gradients of chemical attractant. For the isotropic case, they have found a simple formula for persistence number in terms of a fixed angle between successive trajectories, by using Monte Carlo computer methods. This formula is derived below analytically, and extended to include an arbitrary distribution of angles. The more easily measured quantity, persistence time, is found for an arbitrary dependence of the trajectory duration probability distribution on direction. We call a trajectory positive if it has a positive z-component of direction, and negative otherwise. In a sequence of trajectories, a set of successive positive trajectories which is directly preceded and followed by negative trajectories is called a positive excursion. Thus the sequence of trajectories defines an alternating sequence of positive and negative excursions. Each excursion (e.g., the ith) is characterized by the number of trajectories in the excursion (NJ and the time duratiqn of the excursion (ti). Persistence number m is defined by Macnab & Koshland (1973) as the mean number of trajectories in a positive excursion. Persistence time is defined by them as the mean time duration of a positive excursion. We will call this the positive persistence time, t,, and define negative persistence time, t-, as the mean duration of a negative excursion. We will calculate persistence number for the isotropic case, using the following lemma: the persistence number 1p is the reciprocal of the probability P that one turn will change the sign of a randomly chosen positive trajectory. The lemma can be discussed rigorously by applying certain general theorems relating to stationary ergodic processes (Cramer & Leadbetter, 1967). The brief proof below is heuristic in the simple form given, but is more meaningful intuitively. Letting the even subscripts refer to upward excursions, we have
P = lim
,,--rm N,+N,+N,+.
n . . +Nzn
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and
Thus we can find.fl by tkling P. First we will Cud this probability for a tkced turn angle $, and call the probability p(e); then we will generalize to a distribution of $; and finally we will find t+ and t-.
Consider two sequential trajectories. Let a, b, and z be unit vectors in the initial, final and z directions, respectively. We characterize a trajectory by the cosine of its angle with respect to the z aixis: u =a*z=cos& u’ = b-z = cm 8’. Also, we define w = a *b = cos $, where e is the turn angle, assumed constant for now. The quantities u, u’ and w can range from - 1 to 1. (B) bETHOD
To fmd p(e) in a situation wheref(u) du is the probability that a randomly chosen positive trajectory is in the interval du at u, we let K(u’lu) du’ be the probability of turning into du’ at u’ in one turn from u. Then the probability that the new direction is negative is
so p = 1 duf(u) 4, du’K(u+).
For an isotropic distributionflu)
(16)
= 1, and thus
p = d du j, du’K(u’lu).
To find the function K, picture vectors a, b and z projecting from the origin to the surface of the unit sphere. Draw vector A tangent to the sphere from the tip of a to an intercept on the z axis, and vector B tangentially from the tip of a, intercepting an extension of b. Let 4 be the magnitude of the angle between vectors A and B. It ranges from 0 to II (in two directions) with uniform probability density by assumption i(c). Therefore d4/lc is the probability that 4 is in any interval of magnitude d&
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Given u and W, C#Iand u’ are in a one-to-one correspondence. Thus to find K(u’lu), we need only to find this correspondence and make the statement K(+)
du’ = !i?! = ? !?i du’ ?T ~at.4~ I I
or
W/u)
'
= - ; tg (u’, u),
(18)
since this partial derivative is negative. Because K is a derivative, we can partially evaluate the integral for p before stating the explicit dependence of the function 4 on u’ and u. Let us assume w > 0, or $ < 742. (The other case will be treated by a symmetry argument at the end.) Then for a given u > 0, K(u’lu) will be zero for u’ < ub, where u’ = uh gives C$= n. (ub depends on u). Also, K(u’lu) is zero for all u’ < 0 when u > uO, where u = u,, gives U’ = 0 when 4 = z This defines the effective limits on the integral (17) for p: p = 1 du “I du’K(u’lu). WI’
Substituting equation (18), integrating and noting that by definition
over u’, integrating
by parts over u,
M4, u>= w-4uo> = 71, transforms this to the form
(C) EVALUATION
Now we find q5 and evaluate the integral. As defined above, the vectors A and B are Ad-a,
&La,
U
W
and 4 satisfies
from the definitions of u, u’ and w. At u’ = 0, U ~0s
4
=
- k (1
-u2j+s
(21)
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Differentiating
k equation (21),
AND’
CHEMOTAXIS
W c------x+ (l-w2)*
1 tan $’
487
(22)
-sin 4 W du = -k(l-u2)-3/2. Using cos’ I$ + sin2 4 = 1 and substituting
d4 du =
equation (21) gives
ks-‘[(1+k2)s-k2]-*,
where s = l-u2. Expressing equation (20) in terms of this new variable, we have lk l-U&$ =--s-‘[(l+k2)s-k2]-*ds, P I n2
1
which can be evaluated with tables:
The upper limit leaving
can be seen to vanish by using equations (19) and (21), P
which, by equation (22), reduces to
This formula is also valid for IJ?> n/2. To see this, we note that the probability of changing sign is one minus the probability of not changing sign. But the probability of not changing sign with angle $ equals the probability of changing sign with angle x- $, because turning from a to b with angle $ has the same probability distribution for b as does turning from -atobwithanglerc-+.Thus.
where we have used equation (23), noting that IC- 1,5< 742. If all turns are governed by a probability density W($) such that W(e) d# is the probability that a turn angle is in the interval dt,4 at t,G,then the total probability of changing sign in one turn from a randomly chosen trajectory
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is clearly P = j P(WW
W = i i HW)
W.
The last integral is just lphe average turn angle, which we call $. ‘I&us P = -, v? R and so NC-n 3;’ This agrees with the formula postulated by Macnab & Koshland (1973) for aJixed turn angle $, and demonstrates that their formula is exact even when the angle is variable, provided its mean value is used. The expression (24) for R applies during chemotaxis as well as in the isotropic case, for persistence number depends only on the sequence of directions. By assumption (d), this is independent of other information. (D)
PERSISTENCE
TIME
As with persistence number, rigorous theorems (Cramer & Leadbetter, 1967) can be used to show that the persistence time equals the reciprocal of a quantity which can be expressed as an integral very similar to equation (16). The result, however, follows quickly from another heuristic argument. The positive persistence time is tz-tt,+t,+. . . +t,, __ t+ = lim n “do3 But for large n, t,+t,+t,+. . . +fz” N (Nz+N.$+N6+. . . +N,,)T+, where T+ is the mean duration of a positive trajectory, averaged over all positive directions with equal solid angles weighted equally. From this it follows that t, = NT+. This applies for any choice of the positive z direction. For the purpose of chemotaxis studies, it is convenient to let this direction correspond to the direction of increasing chemical attractant or repellent concentration. Then the problem has axial symmetry, and to express the mean duration of a trajectory as a function of direction, we need only to state its dependence on U, the cosine of the angle of a trajectory with respect to the direction of the gradient. If T(u) is this mean duration, then T+ = i T(u) du.
(25)
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We also define T, = jl T(u) du.
(26)
Then
t, = k, J;
(27)
n-T-.
t- =q
6. Average Velocity During chemotaxis, bacteria drift up an attractant gradient (or down a repellent gradient) with a net average velocity, which can be determined from the distribution of bacteria over all directions of motion. To Grid the steady state direction distribution for an ensemble of identical bacteria, we need only to state that the rate of turning out of any set of trajectory directions equals the rate of turning in. Choosing the co-ordinate directions as was done in the persistence time section, let n(u) + be the number of bacteria with trajectory directions a such that a.2 is in the interval da at 24.Then the rate of turning out of this direction region is given by n(u) du T(u) where T(u) is the mean duration, defined before. Since the geometry of the turning proces$ is isotropic by assumptions (c) and (d), the steady state condition mentioned above will be satisfied if the rate of turning out of any solid angle equals the rate of turning out of any other equal solid angle. Since the solid angle defined by du at u is directly proportional to du (equal to 2a du, independent of u), the condition is met if -n(u) = constant. T(u) We will normalize n(u) so that its integral over all directions is unity, i.e., so that it is the probability density function for the direction cosine, u: 44
T(u) . = 1 j T(u) die
(28)
-1
This defines the steady state velocity distribution during chemotaxis. Because of axial symmetry, the average vector velocity lies along the 2 axis,
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so we need only to find the z component. For a velocity ua, the z component is va. z = vu. Thus the average velocity, denoted by VA, is j T(u)u du 1 VA = j n(u)vu du = v =+ . -1 j T(u) du
(2%
-1
Note that by equations (25) and (26) the denominator
7. Up/Down
is just T- +T+.
Ratio
If freely swimming bacteria are viewed through a microscope with a single crosshair across the field of view, an observer can count the numbers of bacteria which cross the line in both directions. These numbers will depend on the time period of the observations; on the width and depth of the field of view; and on the concentration, speed and direction distribution of the bacteria. But their ratio will depend only on the direction distribution. If both the crosshair and the direction of view are perpendicular to the direction of a chemical gradient, such an experiment gives the ratio of the rate of flow of bacteria traveling up the gradient to the rate of flow of bacteria traveling down the gradient. We call this the up/down ratio, R. It has proved to be a very practical assay of the strength of chemotactic response (Dahlquist & Lovely, 1974). To relate the up/down ratio to the mean duration T(u), we note that the rate, per unit area and time, of bacteria in the direction cosine interval du at u crossing a plane perpendicular to the z axis, is proportional to n(u) vu du. We need to integrate u from 0 to 1 to find the total rate for upward moving bacteria, and from 0 to - 1 for those moving downward. Using equation (28), the up/down ratio is therefore j T(u)u du = R “-1 1 T(u)u du’
8. Chemotaxis
(30)
Parametrization
During chemotaxis, the mean duration T(u) will depend not only on the direction cosine u, but also on the concentration c of chemical attractant, and on its gradient, dc/dz. But recent experiments demonstrate that the mean duration can be expressed as a function of just two variables rather
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than these three. Bacteria respond to temporal changes in c (Macnab & Koshland, 1972), and the mean duration is modulated by the time derivative of c in a way which can quantitatively explain chemotaxis (Brown & Berg, 1974). We can therefore express Z’(U) as a function of just c and its time derivative, uu(dc/dz), as seen by a moving bacterium. But evidence that bacteria respond chiefly to proportional changes (Dahlquist et al., 1972; Mesibov, Ordal & Alder, 1973) suggests that it is more convenient to express T(U) as a function of c and uuy, where y = (dc/dz)/c is the gradient of the logarithm of concentration. Then the c dependence will presumably be weak, and the mean duration will be chiefly a function of one variable. Suppressing the speed v and the weak c dependence, we can write T(u) = F(yu).
(31)
With this parameterization, the dependence of statistical measures (such as V,, t+ and R) on the concentration and gradient of chemical attractant can be discussed in terms of a single biologically interesting function, F. It would be useful if F could be found from the concentration and gradient dependence of some measure of chemotactic ,response. This is the case for persistence time. Substitution of equation (31) in (25) using (27), rearrangement, and differentiation with respect to y, yields
If the multiplicative $/ln is not known, this still determines the shape of the function. Concentration is assumed fixed here, while y varies. The c dependence of F can be found by measuring the y dependence of t+ at different values of c. Average velocity by itself does not tell so much about F as does t+, but certain available additional information makes it comparably informative. Let us express F as the sum of its even and odd parts, F(x) = G(x)+H(x), where 2G(x) = F(x) + F( -x) and 2H(x) = F(x) - F( -x). Then substitution of equation (31) in (29) demonstrates that the numerator of (29) depends only on H, and the denominator only on G. Suppose we know u and have measured the function f(y) = VJu for y between zero and some maximum value. The information in equation (29) can easily be re-expressed as rf(r> / G(x) dx = ,i xH(x) dx from which it is clear that either one of the two functions G and H may be chosen arbitrarily, and knowledge off(y) then determines the other uniquely. For example, to find H when G is given, differentiation of (33) and
492 rearrangement
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yields
So far, this amounts to quite a lot of freedom in finding a function F which predicts an observed dependence of average velocity on gradient. However, it is known that behavior while moving down an attractant gradient is essentially unmodified from behavior in the absence of a gradient (Berg & Brown, 1972), and thus F is constant for negative values of its argument. Calling the constant T, this implies that for positive y, H(y) = G(r) - T and F(y) = 2G(y)-T. Using this new information to express G and H in terms of Fin equation (34), and defining
Q(r) = $ j F(x) dx,
(35)
0
we have, after rearrangement,
(Y-g)Q'=dQ
= (w)‘+y where g(y) = YJ(~) and prime (‘) means differentiation. differential equation is
' rxdx)l' +x -h(X)dx, Q(Y) = ehty'1 X-~(X) e
The solution to this (36)
0
where
From equation (35),
and thus knowledge of II and of the gradient dependence of V, determines F uniquely, apart from the multiplicative factor T. As with persistence time, the c dependence of F can be found by measuring the y dependence of VA
at different values of c. Knowledge of the gradient dependence of the up/down ratio R, like knowledge of VA/v, can determine either G(y) or H(y) once the other is given. Or, equivalently, it determines either F(y) for positive y or F(y) for negative y, once the other is given. Substituting F(p) for T(u) in equation (30) and assuming as above that F(x) = T for negative x; then rearranging and differentiating, we have
F(Y)= -; & bw.
(37)
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These quations demonstrate that the shape of F is uniquely determined by persistence time, average velocity, or up/down ratio measurements; but other methods, such as adjusting parameters in manassumed functional form of F, appear more practical than equations (32), (36) or (37) for actually finding the function from experimental data. 9. -on Equations (3), (14!, (27), (29) and (30) relate the direction correlation time, diffusion constant, persistence time, average velocity, and up/down ratio to the mean angle $, the mean cosine a, the mean isotropic duration T, the mean upward and downward durations T+ and T,, and integrals of UT(U). These results are summarized in Table 1. While the model which relates these quantities is an idealization, it appears to have all the most essential features of the real case. These relations allow the interpretation of changes in statistical behavior (due to mutation, for example) in terms of microscopic behavior. All the statistical measures discussed in this paper for three-dimensional motion have meaning in two dimensions as well. They can be described in terms of a model identical to the one above, except that assumption (c) is replaced by equal probability of turning left or right. Two-dimensional persistence number has been discussed quantitatively by Macnab & Koshland (1973), and the other measures can be found by simple modifications of the mathematics in this paper. Two-dimensional average velocity and diffusion are also described by Nossal & Weiss (1974). Direction correlation time and diffusion constant are simply related by equation (15) when assumption (f) is obeyed. Ewen without that assumption, equation (13) would allow D to be calculated from microscopic tracking data. Hand calculation from the distribution af angles in Fig. 3 of Berg & Brown (1972) yields a N O-33, and their Table 1 gives T = O-86 set, u = 14-2 pm/set, from which (3) and (14) imply TV N l-3 set and D N 87 (pm)2/sec N 04lO3 cm2/hr for wild type Escherichiu co4 AW 405. This is in sharp contrast to the (equivalent) motility coefficient of O-25 cm2/hr N 7000 (pm)2/sec reported by Adler & Dahl (1967) for another strain of E. coli. However, that value applied only to the most rapidly diffusing bacteria, being based on the wings of several very non-gaussian diffusion patterns. Extrapolation of those wings to the center of the pattern, assuming a gaussian curve is correct for those bacteria on which the measurements were based, indicates that the rapidly diffusing bacturia represent only about lo-’ of the total population. The published data demonstrate clearly that the majority of the bacteria diffuse much more slowly. The value of D = 87
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(pm)Z/sec found above is based on a sample of 35 bacteria. Since population averaging is done before calculating D from T, v and a, it is not a true population average of D; but the value should still be quite representative. An estimate of the rotational diffusion contribution decreases z, and D by about 20%, and the non-zero duration of a twiddle decreases D again by 14x, resulting in D N 60 (pm)2/sec N OXlO2 cm2/hr. A preliminary observation of a displacement direction correlation function has been made (Dahlquist & Lovely, 1974) from a 29 min track of a single bacterium, Salmonella typhimurium, using a tracking machine described elsewhere (Lovely et al., 1974). Position was recorded every 5 set, and the result was a good exponential with rC = IO sec. The amplitude was 1 to within experimental error. The mean velocity was about 20 pm/set. From equation (15), this gives D N 1300 (pm)‘/sec N 0448 cm2,1hr. D has also been estimated directly (Dahlquist & Lovely, 1974) for the same bacteria by observing the gradual broadening of a step change in bacteria concentration using a device described elsewhere (Dahlquist et al., 1972). Several repeated measurements indicate that the population average of D% is between 20 and 40 pm/set%, or that D is between 400 and 1600 (pm)2/sec, consistent with the value predicted from the correlation time and velocity of a single bacterium. These values for S. typhimurium are smaller than the motility coefficient of 0.16 cm2/hr N 4400 (pm)2/sec calculated by Nossal & Weiss (1973), or the value 0.2 cm2/hr N 5600 (pm)2/sec calculated by Segel & Jackson (1973), from chemotaxis data of Dahlquist et al. (1973). The disagreement is not alarming, for two reasons. First, the calculation is based on a sample of bacteria enriched for those with a higher average velocity during chemotaxis. Second, diffusion during chemotaxis is not the same as in the isotropic case. In fact, without isotropy it must be described (except in special cases) by a second rank tensor rather than a scalar diffusion constant. Although it is a complicated problem, anisotropic diffusion can be discussed analytically (Lovely, 1974) for the model described in this paper. Persistence time has been measured and discussed quantitatively in the isotropic case by Macnab & Koshland (1973). As equation (32) indicates, measurements made during chemotaxis would give very useful information. Both positive and negative persistence time could be measured simultaneously, and presumably negative persistence time would be the same as in the absence of a gradient, since, as mentioned in the previous section, behavior while traveling down the gradient is the same as for the isotropic case. Both average velocity (Dahlquist et al., 1972; Dahlquist & Lovely, 1974) and up/down ratio (Dahlquist & Lovely, 1974) can be measured accurately
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and reproducibly, and have been studied in various chemical gradients. The gradient dependences of some of these measurements can be fitted to a simple model for the function F (Dahlquist & Lovely, 1974). Perhaps the most crucial feature of the model described in this paper is that the bacteria have no memory of previous events. This is implicit in assumptions (d), (e) and (f). It has been observed (Macnab t Koshland, 1972; Brown & Berg, 1974) that bacteria can “remember” a sudden increase in attractant concentration for a time of the order of a minute or longer. But it is not inconsistent with those observations to suppose that for slower temporal changes such as those a bacterium normally creates by swimming in a smooth gradient, the rate of turning depends on the instantaneous time derivative of attractant concentration seen by the bacterium. (More reasonably, it depends on an average of the time derivative over a period brief compared to a mean trajectory duration.) A computer simulation of bacteria with a memory (Brown & Berg, 1974) indicates that it would be possible for a memory time comparable to the mean isotropic trajectory duration to exist without destroying the effect of chemotaxis. Should the actual memory time be that long, equation (29) would be valid only as an approximation. Assumptions (a)-(f) are extremely useful for microscopic interpretation of the statistical measures described in this paper. Nevertheless, a great deal of information about the molecular mechanisms controlling motility and chemotaxis probably resides in the departures of real bacterial behavior from these idealizations, particularly in the departures from (d), (e) and (f). It is interesting to ask, for example, how much (if any) residual memory exists of changes in attractant concentration prior to the most recent twiddle; or, if further microscopic tracking experiments still fail to find any correlation of behavior with events prior to the most recent twiddle, whether any “induction period” can be measured at the start of a run, before the behavior mechanism responds to the new time derivative of attractant concentration. If, as present data indicate, the departures are not too great, it appears that the statistical measures discussed here may not be very sensitive for studying those departures; but as a result of such insensitivity, these measures will remain particularly useful because theory such as presented here allows their meaning to be interpreted clearly. This work was supported in part by grants GM 00715 and GM 18994 from the U.S. Public Health Service. RBF’BRBNCES AD-
J. C DAHL,
M.
M. (1967).
J. gen.
h&rob&Z.
Emto, H. C. (1971). Rev. Sci. Znstrum. 42,868. BBRO, H. C. & BROWN, D. A. (1972). Nature, Lmd.
46, 161. 239,500.
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BROWN, D. A. &BERG, H. C. (1974). Proc. natn. Acud. CHANDRAsEKHAR, S. (1943). Rev. mod. Phys. 15, 1. CRA&R, H. & LEADBETTER, M. R. (1967). Stationary
Sci. U.S.A. 71, 1388.
and Related Stochastic Processes. New York: Wiley. DAHLQIJIST, F. W. & LOVELY, P. S. (1974). (In preparation.) DAHLQUST, F. W., LoVELY, P. S. & KOSHLAND, JR, D. E. (1972). Nature New Biol. 236, 120. FLORY, P. J. (1969). Statistical Mechanics of Chain Molecules. New York: Wiley. Loamy, P. S. (1974). (In preparation.) LOVELY, P. S., DAHLQUIST, F. W., MACNAB, R. & KOSHLAND, JR, D. E. (1974). Rev. Sci. Instrum. 45,683. MACNAB, R. & KOSHLAND, JR, D. E. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 2509. MACNAB, R. & KOSHLAND, JR, D. E. (1973). J. Mechanochem. Cell Motility 2, 141. MESIBOV, R., ORDAL, G. W. & ADLER, J. (1973). J. gen. Physiol. 62,203. NOSSAL, R. & WEN, G. H. (1973). J. theor. Biol. 41, 143. NOSSAL, R. & WEISS, G. H. (1974). J. theor. Biol. 47, 103. PATLAK, C. S. (1953). Bull. math. Biophys. 15, 311. SEOEL, L. A. & JACKSON, J. L. (1973). J. Mechanochem. Cell Motility 2, 25.
Statistical Measures of Bacterial Motility
and Chemotaxis
PJITER s. LOVELY Department of Physics, and Lwtitute of Molecular Biology, University of Oregon, Eugene, Oregon 97403, U.S.A.
F. W. DAHLQUIST Department of Chemistry, and Institute qf Molecular Biology, University of Oregon, Eugene, Oregon 97403, U.S.A. (Received 23 April 1974, and in revisedfirm
12 August 1974)
This paper presents a model for the three-dimensional microscopic behavior of motile bacteria, and relates its phrameters to five practical measures of bacterial motility and chemotti (direction correlation function, diffusion constant, persistence time, average velocity, and up/down ratio). The attractant gradient depen@encesof persistence time, average velocity, and up/down ratio are reMed to a single function describing the gradient sensing mechanism. 1. Introduction
Much interest in recent literature has been devoted to physical measurements of the behavior of motile bacteria (Adler & D&l, 1967; Dahlquist, Lovely & Koshland, 1972; Berg & Brown, 1972; Macnab & Koshland, 1973), with the aim of understanding the molecular mechanisms of motihty,.especially during chemotaxis. An experiment is more interesting if it yields detailed microscopic information, such as can be found using the tracking techniques of Berg (1971). But experiments done with less detailed tracking (Lovely, Dahlquist, Macnab & Koshland, 1974) or with macroscopic measurement of concentration of bacteria at different positions in a solution (Dahlquist et al., 1972), can still yield valuable microscopic information, provided the results of these experiments can be related mathematically to a detailed microscopic description of bacterial behavior. It appears worthwhile to relate certain measurable statistical quantities to the parameters of a model for bacterial motion suggested by the tracking data of Berg P Brown (1972) for Escheriehika coli. This model, described below, is not very restrictive, in that it leaves the probability distribution for turning angles and the direction dependence of 477 31 T.B.
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the mean trajectory duration unspecified. It is merely a logical idealization of the experimental evidence, which simplifies analysis, without, one hopes, suppressing any of the crucial features. 2. The Model Let us approximate the three-dimensional motion of a bacterium by making the following assumptions : (a) The path of a bacterium is a sequence of straight line trajectories joined by instantaneous turns, each trajectory being characterized by a speed, direction, and time duration. (b) All trajectories have the same constant speed. (c) When a bacterium turns, its choice of a new direction is governed by a probability distribution which is azimuthally symmetric about the initial direction. (d) The angle between two successive trajectories is governed by a probability distribution which is independent of any other information, such as the direction or duration of the initial trajectory. (e) The duration of a trajectory is governed by a probability distribution which may depend on the direction of that trajectory, but on no other information, such as the parameters of previous trajectories. (f) For any particular direction, the probability density function for the duration of a trajectory is a decaying exponential. Assumptions (a) and (b) are not perfect, and their degree of validity may be judged from the data of Berg & Brown (1972). In most of what follows, trajectories and turns correspond respectively to “runs” and “twiddles”, as defined quantitatively by Berg & Brown (1972). However, as is shown below, rotational diffusion during runs can be accommodated by this model when there is no chemotaxis, by letting a trajectory be much shorter than a run. Also, the non-zero duration of a twiddle can be included in the calculation of the diffusion constant. The data of Berg & Brown (1972) are consistent with assumption (c), and support (d), (e) and (f). They include measurements of a distribution for the angle between successive trajectories, and of a nearly exponential distribution of trajectory durations. We will not need assumption (f) except for discussing the direction correlation function. This assumption is equivalent to saying that the probability of turning in any small time interval depends only on the direction of travel at that time, and not on what happens at any other time. For the isotropic case it means that the number of turns in any time interval is governed by a Poisson distribution. For discussing persistence time, average
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velocity and up/down ratio, we need only to assume that the first moment (mean duration) of the duration probability function exists. For diffusion we must require that the second moment (mean square duration) exists also. Chemotaxis arises from the dependence of’ mean trajectory duration on direction in assumption (e). While the data shaw that this dependence exists, they are insufficient to describe it in detail. It is shown below how persistence time, average velocity and up/down ratio can give information on this dependence. Statistical measures which depend on large numbers of bacteria are complicated by the fact that all the bacteria are not identical. Even though each bacterium may obey this model, they will hot all have the same speed or other parameters. To describe the behavior of an ensemble of independent bacteria with different properties is straightforward and will not be discussed here; but the problem must be borne in mind when interpreting experiments. In terms of this model, five different statistical measures, each of which has been studied experimentally, are described in detail below. The results are summarized in the Discussion and in Table 1. TABLE 1 Summary of formulae relating statistical measures to microscopic behavior information
t+ =$T(u)du=;;jF(x)dx
R = j UT(U) du j xF(x) dx = -1 - 7. J UT(U) du I0 xF(x) dx 0
Correlationtime T.. diffusionconstantD, persistence time t+, averagevelocity VA and up/downratio R are relatedto the meanisotropicqajectory duration T, the direction dependent meandurationT(u), the speedO,the meanturn angle4, andthe meancosinea. The dependence of t+, VA and R on the proportional&tractant gradienty isexpressed in termsof the function F describingthe dependence of meantrajectory duration on the attmctant concentrationand its time derivative.(The concentrationdependence of F is not shown.) 310
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3. Direction Correlation Function We characterize the instantaneous direction of motion of a bacterium by a time-dependent unit vector a(z). We define the direction correlation function by CO> = W)-W>, where < ) means the expectation, averaging over all initial directions a(O), and all possible subsequent paths. Although C(t) can have meaning during chemotaxis, we consider here only cases where all directions are equivalent. Clearly we do not need then to average over initial directions. We must average over the number of trajectories between zero and t; over the angles between these trajectories; and over the azimuthal angles. We do these in reverse order. Suppose n turns occur between 0 and t, and that the n+ 1 trajectories they link have directions a,, al, . . ., a,. Let a,*a, = CI,, a, *a, = CQ, . . ., an-l*an = ~1,. Then, denoting azimuthal angle averaging by ( )+, we have (a(O)*a(t))+, = (a0*a,)4
= alclz . . . a,.
Here we have used an argument familiar in polymer chemistry (Flory, 1969). Averaging now over angles between trajectories (denoted by a subscript $ on the brackets) we have G@)MO>,,
Q = a”,
(1)
where a is the mean cosine of the angle between successive trajectories. This follows because the angles are independent. Finally, averaging over the number of turns which occur between 0 and t, we have C(t) = P,a”+P,a’+P,a2+.
. . +P,a”+.
. .,
where P, is the probability that n turns occur in time t. This sum does not have a simple analytic form in general; but for the specific case of Poisson statistics, Pn has the form p _ (tP)” n n!
e-t/T ’
and the sum is easy to carry out, since ~ ifi (t’T)“a” n=o n!
= ew
This yields C(t) = esf”c,
(2)
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where the correlation
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time is T zc = (1-a)
(3)
Here T is the mean duration of a trajectory. The formula (2) applies to positive t. To include negative, t should be replaced by It 1. It can also be derived simply by starting with the Poisson assumption and stating that C(t + At) = C(t)C(At), which yields a differential equation. We use the above method because ‘it illustrates the complicated nature of the problem in the absence of Poisson statistics. Equation (2) can accommodate the rotational diffusion during runs observed by Berg & Brown (1972). Instead of considering a run to be a single straight line trajectory, we can describe it as a sequence of many very short trajectories. The Poisson process described by T and a may be considered as two simultaneous independent Poisson processes, one of largeangle turns (twiddles) with mean cosine a, at a slow rate l/T,, and another process of small-angle turns (rotational diffusion) with mean cosine cr, at a rapid rate l/T,. Using (3), it is easy to deduce that for this combined process 1 -=A,1 (4) 7, *t 2,’ where 2 T, 5 = G-cr,’ Tt =i--at The “relaxation rates” add in equation (4), as we might expect intuitively for two independent processes. In practical cases, T, and 1 -a, might both be itinitesimal, so that r, is the only meaningful parameter for rotational diffusion. It is worth noting here that we would be unable to describe rotational diffusion during chemotaxis in terms of the model presented in this paper. If one turning rate is isotropic and the other is not, assumption (d) is violated. The direction correlation function can be measured by tracking bacteria with less precision than would be required to record a(t) in detail, because if the assumptions (a)-(f) are obeyed, it is related to another function, the displacement direction correlation function, defined below. Suppose we cannot measure the instantaneous position of a bacterium with enough resolution in time and space to irecord a(r), but that we can record a rough position at regular time intervals, where the interval r between measurements is perhaps comparable to the correlation time. We assume our position measurements are good enough to give a reasonably accurate measurement of the vector displacement between two position measurements.
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We define the displacement direction b(t,, tz) to be a unit vector pointing from the position at time t, to the position at time tz. And we define the displacement direction correlation function to be C,(t) = (b(0, z).b(t, t+z)>. We will discuss only the case where t > z, so that the time intervals (0, r) and (t, t+ T) are disjoint. Azimuthal angle symmetry then insures that components of h(0, r) perpendicular to the final direction a(r) average to zero, as do components of b(t, t+ z) perpendicular to the initial direction a(t). Saving only the parallel components, we can write
C,tO= (MO, 3 *at+(~)] -[atOW>-btt, t + $I> = (WA ~).at~))(a(~).atr))(a(~).b(r,t+4>,
(5) since the three quantities are uncorrelated. The factor in the middle is
N */3> 1+(xo2/
- l)a
1-rx ’ where u now denotes the mean projection of one bond direction on the next. This hypothetical polymer example superimposes on our model for bacterial motion. Replacing n by t/T where T is the mean duration of a trajectory, and interpreting a as in (l), (11) combined with (12) yields D _ 02> 1 +w>2/(~2~ - 1P , l-a 6T where (2’) is the mean square distance traveled between turns, and (I) is the mean distance. This equation can also be #found using the random walk treatment of Patlak (1953). If each new direction is chosen at random, tc = 0 and this reduces to a familiar expression for D in terms of mean square free path and flight time (Chandrasekhar, 1943). For the case where the speed is constant, the formula reduces to D U’S 1+(2T”/S+ 1)cr =(13) 6T l-a ’ where S is the mean square duration of a trajectory, and u is the speed. If assumption (f) holds, it is easy to show that S = 2T2, so the diffusion constant takes the form v2T D (14) = giq*
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Note that by equation (3) we have
D = +=zc,
(15)
which holds even when rotational diffusion is included, as described before. The finite duration of a twiddle can be accommodated by assuming that zero distance is traversed during that time, and replacing n in (12) by t/(T+T,) rather than t/T, where Tl is the mean twiddle duration. This multiplies the expression (14) by T/(T+ Tl). 5. Persistence Time
Macnab & Koshland (1973) have defined persistence time and persistence number, and made measurements of persistence time, in the presence and absence of gradients of chemical attractant. For the isotropic case, they have found a simple formula for persistence number in terms of a fixed angle between successive trajectories, by using Monte Carlo computer methods. This formula is derived below analytically, and extended to include an arbitrary distribution of angles. The more easily measured quantity, persistence time, is found for an arbitrary dependence of the trajectory duration probability distribution on direction. We call a trajectory positive if it has a positive z-component of direction, and negative otherwise. In a sequence of trajectories, a set of successive positive trajectories which is directly preceded and followed by negative trajectories is called a positive excursion. Thus the sequence of trajectories defines an alternating sequence of positive and negative excursions. Each excursion (e.g., the ith) is characterized by the number of trajectories in the excursion (NJ and the time duratiqn of the excursion (ti). Persistence number m is defined by Macnab & Koshland (1973) as the mean number of trajectories in a positive excursion. Persistence time is defined by them as the mean time duration of a positive excursion. We will call this the positive persistence time, t,, and define negative persistence time, t-, as the mean duration of a negative excursion. We will calculate persistence number for the isotropic case, using the following lemma: the persistence number 1p is the reciprocal of the probability P that one turn will change the sign of a randomly chosen positive trajectory. The lemma can be discussed rigorously by applying certain general theorems relating to stationary ergodic processes (Cramer & Leadbetter, 1967). The brief proof below is heuristic in the simple form given, but is more meaningful intuitively. Letting the even subscripts refer to upward excursions, we have
P = lim
,,--rm N,+N,+N,+.
n . . +Nzn
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and
Thus we can find.fl by tkling P. First we will Cud this probability for a tkced turn angle $, and call the probability p(e); then we will generalize to a distribution of $; and finally we will find t+ and t-.
Consider two sequential trajectories. Let a, b, and z be unit vectors in the initial, final and z directions, respectively. We characterize a trajectory by the cosine of its angle with respect to the z aixis: u =a*z=cos& u’ = b-z = cm 8’. Also, we define w = a *b = cos $, where e is the turn angle, assumed constant for now. The quantities u, u’ and w can range from - 1 to 1. (B) bETHOD
To fmd p(e) in a situation wheref(u) du is the probability that a randomly chosen positive trajectory is in the interval du at u, we let K(u’lu) du’ be the probability of turning into du’ at u’ in one turn from u. Then the probability that the new direction is negative is
so p = 1 duf(u) 4, du’K(u+).
For an isotropic distributionflu)
(16)
= 1, and thus
p = d du j, du’K(u’lu).
To find the function K, picture vectors a, b and z projecting from the origin to the surface of the unit sphere. Draw vector A tangent to the sphere from the tip of a to an intercept on the z axis, and vector B tangentially from the tip of a, intercepting an extension of b. Let 4 be the magnitude of the angle between vectors A and B. It ranges from 0 to II (in two directions) with uniform probability density by assumption i(c). Therefore d4/lc is the probability that 4 is in any interval of magnitude d&
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Given u and W, C#Iand u’ are in a one-to-one correspondence. Thus to find K(u’lu), we need only to find this correspondence and make the statement K(+)
du’ = !i?! = ? !?i du’ ?T ~at.4~ I I
or
W/u)
'
= - ; tg (u’, u),
(18)
since this partial derivative is negative. Because K is a derivative, we can partially evaluate the integral for p before stating the explicit dependence of the function 4 on u’ and u. Let us assume w > 0, or $ < 742. (The other case will be treated by a symmetry argument at the end.) Then for a given u > 0, K(u’lu) will be zero for u’ < ub, where u’ = uh gives C$= n. (ub depends on u). Also, K(u’lu) is zero for all u’ < 0 when u > uO, where u = u,, gives U’ = 0 when 4 = z This defines the effective limits on the integral (17) for p: p = 1 du “I du’K(u’lu). WI’
Substituting equation (18), integrating and noting that by definition
over u’, integrating
by parts over u,
M4, u>= w-4uo> = 71, transforms this to the form
(C) EVALUATION
Now we find q5 and evaluate the integral. As defined above, the vectors A and B are Ad-a,
&La,
U
W
and 4 satisfies
from the definitions of u, u’ and w. At u’ = 0, U ~0s
4
=
- k (1
-u2j+s
(21)
MOTILITY
Differentiating
k equation (21),
AND’
CHEMOTAXIS
W c------x+ (l-w2)*
1 tan $’
487
(22)
-sin 4 W du = -k(l-u2)-3/2. Using cos’ I$ + sin2 4 = 1 and substituting
d4 du =
equation (21) gives
ks-‘[(1+k2)s-k2]-*,
where s = l-u2. Expressing equation (20) in terms of this new variable, we have lk l-U&$ =--s-‘[(l+k2)s-k2]-*ds, P I n2
1
which can be evaluated with tables:
The upper limit leaving
can be seen to vanish by using equations (19) and (21), P
which, by equation (22), reduces to
This formula is also valid for IJ?> n/2. To see this, we note that the probability of changing sign is one minus the probability of not changing sign. But the probability of not changing sign with angle $ equals the probability of changing sign with angle x- $, because turning from a to b with angle $ has the same probability distribution for b as does turning from -atobwithanglerc-+.Thus.
where we have used equation (23), noting that IC- 1,5< 742. If all turns are governed by a probability density W($) such that W(e) d# is the probability that a turn angle is in the interval dt,4 at t,G,then the total probability of changing sign in one turn from a randomly chosen trajectory
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is clearly P = j P(WW
W = i i HW)
W.
The last integral is just lphe average turn angle, which we call $. ‘I&us P = -, v? R and so NC-n 3;’ This agrees with the formula postulated by Macnab & Koshland (1973) for aJixed turn angle $, and demonstrates that their formula is exact even when the angle is variable, provided its mean value is used. The expression (24) for R applies during chemotaxis as well as in the isotropic case, for persistence number depends only on the sequence of directions. By assumption (d), this is independent of other information. (D)
PERSISTENCE
TIME
As with persistence number, rigorous theorems (Cramer & Leadbetter, 1967) can be used to show that the persistence time equals the reciprocal of a quantity which can be expressed as an integral very similar to equation (16). The result, however, follows quickly from another heuristic argument. The positive persistence time is tz-tt,+t,+. . . +t,, __ t+ = lim n “do3 But for large n, t,+t,+t,+. . . +fz” N (Nz+N.$+N6+. . . +N,,)T+, where T+ is the mean duration of a positive trajectory, averaged over all positive directions with equal solid angles weighted equally. From this it follows that t, = NT+. This applies for any choice of the positive z direction. For the purpose of chemotaxis studies, it is convenient to let this direction correspond to the direction of increasing chemical attractant or repellent concentration. Then the problem has axial symmetry, and to express the mean duration of a trajectory as a function of direction, we need only to state its dependence on U, the cosine of the angle of a trajectory with respect to the direction of the gradient. If T(u) is this mean duration, then T+ = i T(u) du.
(25)
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We also define T, = jl T(u) du.
(26)
Then
t, = k, J;
(27)
n-T-.
t- =q
6. Average Velocity During chemotaxis, bacteria drift up an attractant gradient (or down a repellent gradient) with a net average velocity, which can be determined from the distribution of bacteria over all directions of motion. To Grid the steady state direction distribution for an ensemble of identical bacteria, we need only to state that the rate of turning out of any set of trajectory directions equals the rate of turning in. Choosing the co-ordinate directions as was done in the persistence time section, let n(u) + be the number of bacteria with trajectory directions a such that a.2 is in the interval da at 24.Then the rate of turning out of this direction region is given by n(u) du T(u) where T(u) is the mean duration, defined before. Since the geometry of the turning proces$ is isotropic by assumptions (c) and (d), the steady state condition mentioned above will be satisfied if the rate of turning out of any solid angle equals the rate of turning out of any other equal solid angle. Since the solid angle defined by du at u is directly proportional to du (equal to 2a du, independent of u), the condition is met if -n(u) = constant. T(u) We will normalize n(u) so that its integral over all directions is unity, i.e., so that it is the probability density function for the direction cosine, u: 44
T(u) . = 1 j T(u) die
(28)
-1
This defines the steady state velocity distribution during chemotaxis. Because of axial symmetry, the average vector velocity lies along the 2 axis,
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so we need only to find the z component. For a velocity ua, the z component is va. z = vu. Thus the average velocity, denoted by VA, is j T(u)u du 1 VA = j n(u)vu du = v =+ . -1 j T(u) du
(2%
-1
Note that by equations (25) and (26) the denominator
7. Up/Down
is just T- +T+.
Ratio
If freely swimming bacteria are viewed through a microscope with a single crosshair across the field of view, an observer can count the numbers of bacteria which cross the line in both directions. These numbers will depend on the time period of the observations; on the width and depth of the field of view; and on the concentration, speed and direction distribution of the bacteria. But their ratio will depend only on the direction distribution. If both the crosshair and the direction of view are perpendicular to the direction of a chemical gradient, such an experiment gives the ratio of the rate of flow of bacteria traveling up the gradient to the rate of flow of bacteria traveling down the gradient. We call this the up/down ratio, R. It has proved to be a very practical assay of the strength of chemotactic response (Dahlquist & Lovely, 1974). To relate the up/down ratio to the mean duration T(u), we note that the rate, per unit area and time, of bacteria in the direction cosine interval du at u crossing a plane perpendicular to the z axis, is proportional to n(u) vu du. We need to integrate u from 0 to 1 to find the total rate for upward moving bacteria, and from 0 to - 1 for those moving downward. Using equation (28), the up/down ratio is therefore j T(u)u du = R “-1 1 T(u)u du’
8. Chemotaxis
(30)
Parametrization
During chemotaxis, the mean duration T(u) will depend not only on the direction cosine u, but also on the concentration c of chemical attractant, and on its gradient, dc/dz. But recent experiments demonstrate that the mean duration can be expressed as a function of just two variables rather
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than these three. Bacteria respond to temporal changes in c (Macnab & Koshland, 1972), and the mean duration is modulated by the time derivative of c in a way which can quantitatively explain chemotaxis (Brown & Berg, 1974). We can therefore express Z’(U) as a function of just c and its time derivative, uu(dc/dz), as seen by a moving bacterium. But evidence that bacteria respond chiefly to proportional changes (Dahlquist et al., 1972; Mesibov, Ordal & Alder, 1973) suggests that it is more convenient to express T(U) as a function of c and uuy, where y = (dc/dz)/c is the gradient of the logarithm of concentration. Then the c dependence will presumably be weak, and the mean duration will be chiefly a function of one variable. Suppressing the speed v and the weak c dependence, we can write T(u) = F(yu).
(31)
With this parameterization, the dependence of statistical measures (such as V,, t+ and R) on the concentration and gradient of chemical attractant can be discussed in terms of a single biologically interesting function, F. It would be useful if F could be found from the concentration and gradient dependence of some measure of chemotactic ,response. This is the case for persistence time. Substitution of equation (31) in (25) using (27), rearrangement, and differentiation with respect to y, yields
If the multiplicative $/ln is not known, this still determines the shape of the function. Concentration is assumed fixed here, while y varies. The c dependence of F can be found by measuring the y dependence of t+ at different values of c. Average velocity by itself does not tell so much about F as does t+, but certain available additional information makes it comparably informative. Let us express F as the sum of its even and odd parts, F(x) = G(x)+H(x), where 2G(x) = F(x) + F( -x) and 2H(x) = F(x) - F( -x). Then substitution of equation (31) in (29) demonstrates that the numerator of (29) depends only on H, and the denominator only on G. Suppose we know u and have measured the function f(y) = VJu for y between zero and some maximum value. The information in equation (29) can easily be re-expressed as rf(r> / G(x) dx = ,i xH(x) dx from which it is clear that either one of the two functions G and H may be chosen arbitrarily, and knowledge off(y) then determines the other uniquely. For example, to find H when G is given, differentiation of (33) and
492 rearrangement
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So far, this amounts to quite a lot of freedom in finding a function F which predicts an observed dependence of average velocity on gradient. However, it is known that behavior while moving down an attractant gradient is essentially unmodified from behavior in the absence of a gradient (Berg & Brown, 1972), and thus F is constant for negative values of its argument. Calling the constant T, this implies that for positive y, H(y) = G(r) - T and F(y) = 2G(y)-T. Using this new information to express G and H in terms of Fin equation (34), and defining
Q(r) = $ j F(x) dx,
(35)
0
we have, after rearrangement,
(Y-g)Q'=dQ
= (w)‘+y where g(y) = YJ(~) and prime (‘) means differentiation. differential equation is
' rxdx)l' +x -h(X)dx, Q(Y) = ehty'1 X-~(X) e
The solution to this (36)
0
where
From equation (35),
and thus knowledge of II and of the gradient dependence of V, determines F uniquely, apart from the multiplicative factor T. As with persistence time, the c dependence of F can be found by measuring the y dependence of VA
at different values of c. Knowledge of the gradient dependence of the up/down ratio R, like knowledge of VA/v, can determine either G(y) or H(y) once the other is given. Or, equivalently, it determines either F(y) for positive y or F(y) for negative y, once the other is given. Substituting F(p) for T(u) in equation (30) and assuming as above that F(x) = T for negative x; then rearranging and differentiating, we have
F(Y)= -; & bw.
(37)
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These quations demonstrate that the shape of F is uniquely determined by persistence time, average velocity, or up/down ratio measurements; but other methods, such as adjusting parameters in manassumed functional form of F, appear more practical than equations (32), (36) or (37) for actually finding the function from experimental data. 9. -on Equations (3), (14!, (27), (29) and (30) relate the direction correlation time, diffusion constant, persistence time, average velocity, and up/down ratio to the mean angle $, the mean cosine a, the mean isotropic duration T, the mean upward and downward durations T+ and T,, and integrals of UT(U). These results are summarized in Table 1. While the model which relates these quantities is an idealization, it appears to have all the most essential features of the real case. These relations allow the interpretation of changes in statistical behavior (due to mutation, for example) in terms of microscopic behavior. All the statistical measures discussed in this paper for three-dimensional motion have meaning in two dimensions as well. They can be described in terms of a model identical to the one above, except that assumption (c) is replaced by equal probability of turning left or right. Two-dimensional persistence number has been discussed quantitatively by Macnab & Koshland (1973), and the other measures can be found by simple modifications of the mathematics in this paper. Two-dimensional average velocity and diffusion are also described by Nossal & Weiss (1974). Direction correlation time and diffusion constant are simply related by equation (15) when assumption (f) is obeyed. Ewen without that assumption, equation (13) would allow D to be calculated from microscopic tracking data. Hand calculation from the distribution af angles in Fig. 3 of Berg & Brown (1972) yields a N O-33, and their Table 1 gives T = O-86 set, u = 14-2 pm/set, from which (3) and (14) imply TV N l-3 set and D N 87 (pm)2/sec N 04lO3 cm2/hr for wild type Escherichiu co4 AW 405. This is in sharp contrast to the (equivalent) motility coefficient of O-25 cm2/hr N 7000 (pm)2/sec reported by Adler & Dahl (1967) for another strain of E. coli. However, that value applied only to the most rapidly diffusing bacteria, being based on the wings of several very non-gaussian diffusion patterns. Extrapolation of those wings to the center of the pattern, assuming a gaussian curve is correct for those bacteria on which the measurements were based, indicates that the rapidly diffusing bacturia represent only about lo-’ of the total population. The published data demonstrate clearly that the majority of the bacteria diffuse much more slowly. The value of D = 87
494
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(pm)Z/sec found above is based on a sample of 35 bacteria. Since population averaging is done before calculating D from T, v and a, it is not a true population average of D; but the value should still be quite representative. An estimate of the rotational diffusion contribution decreases z, and D by about 20%, and the non-zero duration of a twiddle decreases D again by 14x, resulting in D N 60 (pm)2/sec N OXlO2 cm2/hr. A preliminary observation of a displacement direction correlation function has been made (Dahlquist & Lovely, 1974) from a 29 min track of a single bacterium, Salmonella typhimurium, using a tracking machine described elsewhere (Lovely et al., 1974). Position was recorded every 5 set, and the result was a good exponential with rC = IO sec. The amplitude was 1 to within experimental error. The mean velocity was about 20 pm/set. From equation (15), this gives D N 1300 (pm)‘/sec N 0448 cm2,1hr. D has also been estimated directly (Dahlquist & Lovely, 1974) for the same bacteria by observing the gradual broadening of a step change in bacteria concentration using a device described elsewhere (Dahlquist et al., 1972). Several repeated measurements indicate that the population average of D% is between 20 and 40 pm/set%, or that D is between 400 and 1600 (pm)2/sec, consistent with the value predicted from the correlation time and velocity of a single bacterium. These values for S. typhimurium are smaller than the motility coefficient of 0.16 cm2/hr N 4400 (pm)2/sec calculated by Nossal & Weiss (1973), or the value 0.2 cm2/hr N 5600 (pm)2/sec calculated by Segel & Jackson (1973), from chemotaxis data of Dahlquist et al. (1973). The disagreement is not alarming, for two reasons. First, the calculation is based on a sample of bacteria enriched for those with a higher average velocity during chemotaxis. Second, diffusion during chemotaxis is not the same as in the isotropic case. In fact, without isotropy it must be described (except in special cases) by a second rank tensor rather than a scalar diffusion constant. Although it is a complicated problem, anisotropic diffusion can be discussed analytically (Lovely, 1974) for the model described in this paper. Persistence time has been measured and discussed quantitatively in the isotropic case by Macnab & Koshland (1973). As equation (32) indicates, measurements made during chemotaxis would give very useful information. Both positive and negative persistence time could be measured simultaneously, and presumably negative persistence time would be the same as in the absence of a gradient, since, as mentioned in the previous section, behavior while traveling down the gradient is the same as for the isotropic case. Both average velocity (Dahlquist et al., 1972; Dahlquist & Lovely, 1974) and up/down ratio (Dahlquist & Lovely, 1974) can be measured accurately
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and reproducibly, and have been studied in various chemical gradients. The gradient dependences of some of these measurements can be fitted to a simple model for the function F (Dahlquist & Lovely, 1974). Perhaps the most crucial feature of the model described in this paper is that the bacteria have no memory of previous events. This is implicit in assumptions (d), (e) and (f). It has been observed (Macnab t Koshland, 1972; Brown & Berg, 1974) that bacteria can “remember” a sudden increase in attractant concentration for a time of the order of a minute or longer. But it is not inconsistent with those observations to suppose that for slower temporal changes such as those a bacterium normally creates by swimming in a smooth gradient, the rate of turning depends on the instantaneous time derivative of attractant concentration seen by the bacterium. (More reasonably, it depends on an average of the time derivative over a period brief compared to a mean trajectory duration.) A computer simulation of bacteria with a memory (Brown & Berg, 1974) indicates that it would be possible for a memory time comparable to the mean isotropic trajectory duration to exist without destroying the effect of chemotaxis. Should the actual memory time be that long, equation (29) would be valid only as an approximation. Assumptions (a)-(f) are extremely useful for microscopic interpretation of the statistical measures described in this paper. Nevertheless, a great deal of information about the molecular mechanisms controlling motility and chemotaxis probably resides in the departures of real bacterial behavior from these idealizations, particularly in the departures from (d), (e) and (f). It is interesting to ask, for example, how much (if any) residual memory exists of changes in attractant concentration prior to the most recent twiddle; or, if further microscopic tracking experiments still fail to find any correlation of behavior with events prior to the most recent twiddle, whether any “induction period” can be measured at the start of a run, before the behavior mechanism responds to the new time derivative of attractant concentration. If, as present data indicate, the departures are not too great, it appears that the statistical measures discussed here may not be very sensitive for studying those departures; but as a result of such insensitivity, these measures will remain particularly useful because theory such as presented here allows their meaning to be interpreted clearly. This work was supported in part by grants GM 00715 and GM 18994 from the U.S. Public Health Service. RBF’BRBNCES AD-
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and Related Stochastic Processes. New York: Wiley. DAHLQIJIST, F. W. & LOVELY, P. S. (1974). (In preparation.) DAHLQUST, F. W., LoVELY, P. S. & KOSHLAND, JR, D. E. (1972). Nature New Biol. 236, 120. FLORY, P. J. (1969). Statistical Mechanics of Chain Molecules. New York: Wiley. Loamy, P. S. (1974). (In preparation.) LOVELY, P. S., DAHLQUIST, F. W., MACNAB, R. & KOSHLAND, JR, D. E. (1974). Rev. Sci. Instrum. 45,683. MACNAB, R. & KOSHLAND, JR, D. E. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 2509. MACNAB, R. & KOSHLAND, JR, D. E. (1973). J. Mechanochem. Cell Motility 2, 141. MESIBOV, R., ORDAL, G. W. & ADLER, J. (1973). J. gen. Physiol. 62,203. NOSSAL, R. & WEN, G. H. (1973). J. theor. Biol. 41, 143. NOSSAL, R. & WEISS, G. H. (1974). J. theor. Biol. 47, 103. PATLAK, C. S. (1953). Bull. math. Biophys. 15, 311. SEOEL, L. A. & JACKSON, J. L. (1973). J. Mechanochem. Cell Motility 2, 25.
