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Analysis of Migrating Bands of Chemotactic Bacteria

Michelle Cobeaga

Chemotaxis is defined as the orientation or movement of an organism or cell in relation to chemical agents [1]. For years, scientists have studied this behavior in certain species of bacteria. Through experimentation, they discovered that when motile bacterial cells are placed at one end of a closed capillary tube filled with a medium containing an energy source capable of supporting movement, it is very visible to see that bacteria move preferentially toward higher concentrations of oxygen, minerals, and organic nutrients. This chemotatic behavior produces a sharp and clearly visible gradually migrating band [2]. This paper studies and follows the theory of the traveling bands as proposed by Evelyn Keller and Lee Segel, in which they use partial differential equations to describe the consumption of nutrients and the change in bacterial density due to diffusion and chemotaxis. Collectively, they determine the development and movement of migrating bands. This paper will also analyze and discuss different experiments performed to study chemotatic behavior and their respective results.

The theory in question is the Keller-Segel Theory. Developed by Evelyn Keller and Lee Segel, it presents a phenomenological model for the temporal development of a migrating bacterial band. They model the chemotaxis from a one-dimensional biased random walk, the nutrients being the force which prompts bacterial motion. This motion can be broken down into two alternating states: a straight run of an average duration, followed by an uncoordinated tumble which randomizes the direction of the next run[3,4]. When analyzing this motion and in setting up the partial differential equations describing the nutrient and bacterial concentrations, several things need to be taken into account that are directly associated with the motion: the chemotatic response of the bacteria, the non-chemotactic or diffusive motion of the bacteria, and the bacterial consumption of the nutrients. Diffusion, in this case, is the process in which bacteria moves from a region where their concentration is high to a region where their concentration is low.

The concentration $s(x,t)$ of the nutrients is modeled by Keller and Segel's equation

\frac{\partial s}{\partial t} = -k(s)b + D\left[
{\partial{x^{2}}} \right]\\
\end{displaymath} (1)

where $k(s)$ is the rate of consumption of the nutrients per cell, $b(x,t)$ is the density of bacteria, $D$ is the diffusion constant of the substrate, $x$ is the distance along the tube, and $t$ is the time. This equation takes into account two important properties of the nutrients. The first term on the right hand side of the equation describes the constant bacterial consumption of the nutrients and the second term models the natural diffusive motion of the nutrients [2]. The outward diffusive motion can be thought of in terms of flux, which is the rate of passage of fluid flow across a unit area. The gradient vector points in the direction of the greatest increase of nutrients. The flux$(\vec j)$ of nutrients through a closed surface is proportional to $-\nabla{s}$, i.e. $\vec j = -D\nabla{s}$:

Flux = \int_V\nabla{(-D\nabla{s})}dV.

The concentration of nutrients in a closed region $V$ is $\int_V s\ dV$. Therefore, the concentration of nutrients is changing according to $\int_V
\frac{\partial s}{\partial t} dV$. In this case,

\int_V \frac{\partial s}{\partial t} dV = \int_V -k(s)b dV - \int_V \nabla{(-D\nabla{s})} dV,

where the negative sign preceding the second term on the right hand side of the equation implies a positive flux, where nutrients are exiting a closed surface. So,

\int_V \frac{\partial s}{\partial t} dV = \int_V (-k(s)b + D \nabla^2{s}) dV,

for every V. Therefore, in one dimension, $\frac{\partial s}{\partial t} = -k(s)b + D\nabla^2{s} = -k(s)b +

Keller and Segel's corresponding equation for the concentration of bacteria is

\frac{\partial b}{\partial t} = \frac{\partial }{\partial{x...
\end{displaymath} (2)

where $\chi(s)$ is the chemotactic coefficient measuring the strength of chemotaxis, $\mu(s)$ is the motility or diffusion coefficient for the bacteria, $s(x,t)$ is the density of nutrients, $x$ is the distance along the tube, and $t$ is the time. In this equation, both terms on the right hand side are defined in terms of bacterial flux. The first term on the right hand side of the equation describes the random diffusive motion of bacteria in the $x$-direction without chemotactic influence. The second term on the right hand side of the equation describes the biased random motion of bacteria in the $x$-direction under the influence of chemotaxis. In this case, the flux is proportional to the density of the bacteria present. In order to analyze and solve these two equations, several boundary conditions must be set, such that $b(x,0)=b_0(x), \
s(x,0)=s_0(x),$ $\partial{s}/\partial{x} = 0,
\partial{b}/\partial{x} = 0$, at $x = 0$ and $x = L$. These conditions conveniently orient the capillary tube along the $x$-axis and insure that neither bacteria nor substrate flow out of the ends of the capillary tube. When $D$ = 0, the two equations are solved by looking for solutions in the form of a travelling band moving without distortion in the direction of increasing $x$, where $b(x,t) = b(\xi),\ s(x,t) = s(\xi),\ \xi = x - ct$. Solving these equations produces equations for $s$, $b$, and for $c$, the speed of the migrating band:
\frac{s}{s_{\infty}} = (1+e^{-\bar \xi})^{-1/(\bar \delta - 1)}
\end{displaymath} (3)

\frac{b}{c^{2}s_{\infty}(\mu k)^{-1}} = \frac{1}{\bar \delt...
...^{-\bar \xi}(1+e^{-\bar\xi})^{-\bar\delta / (\bar \delta -1)}
\end{displaymath} (4)

c = Nk/(as_{\infty})
\end{displaymath} (5)

where $s_{\infty}$ is the initial concentration of nutrients, $\bar \xi = c\xi/\mu, \bar \delta = \delta/\mu$, $N$ is the total number of bacteria in the band, and $a$ is the cross-sectional area of the capillary tube[2].

Much can be learned about the role of chemotaxis on bacterial motion by comparing experimental results to the the results produced by the Keller-Segel model. Using their equation for the speed of migrating bands, Keller and Segel compared speed values of migrating bands obtained from previous experiments to calculated values. In once such comparison, a band was observed to be migrating to higher substrate concentration at 0.9 cm/hr. A speed of 1.5 cm/hr was obtained using the equation. These results are reasonably close, with respect to the crude estimates that had to be made for the values of $N$,$s_{\infty}$, and $k$, showing that the Keller-Segel model provides reliable predictions for band speed[2].

Keller and Segel's equation for substrate concentration (3), along with the measured width of the migrating band, also provided them with a way to estimate $\bar\delta = \delta/\mu$, the ratio of the chemotactic strength to motility. Using proper estimates, they found values of $\bar\delta$ between 1 and 2. These values give a numerical measure of the relative strength of chemotaxis. The values also showed that bands would not appear unless $\delta$ was larger than $\mu$, because the order of motion produced by chemotaxis must be large enough to overcome the disordering motion due to diffusion. Secondly, the values were accurate in that they also predicted the shape of the bacterial bands. If $\bar\delta >
2$, the front edge of the band was steeper than the lagging edge, and if $1<\delta<2$, the lagging edge was steeper. These values were in agreement with observation[2].

Michael Holz and Sow-Hsin Chen also observed migrating bands of chemotactic bacteria. In their experiments, they used a photon correlation spectrometer. This device produced an intensity auto-correlation function and was used so that the development of the bacterial density profile could be observed by the scattered light intensity as the band migrated through a stationary laser beam. Holz and Chen compared the bacterial density profile produced by the photon correlation spectrometer with Keller and Segel's model for travelling bands and made similar comparisons concerning speed. With proper estimates, their calculated value for the speed of the migrating band was 0.9 $\mu$m/s, and the observed speed was 0.7 $\mu$m/s. Their results were relatively accurate. Similarly, they also calculated $\bar\delta$, the ratio of chemotactic strength to motility. For given values of $\mu$ and of the band width, they obtained a ratio $\bar\delta$ = 1.72, compared to the expected ratio of 1.33. They also found that the shape of the band profile was in good agreement with the shape proposed by the model. For a given ratio of 1.33, the calculated curves for bacterial density distribution were very similar to the measured curves predicted by the Keller-Segel model. It also complied with the prediction that for $\bar\delta<2$, the back edge was steeper than the front edge[3]. Altogether, they further showed that the model represented certain features of the migrating band very reasonably. However, there were some discrepancies, most likely due to the assumption of $D$ = 0 in the Keller-Segel model. The density profile is either too broad or the speed of migration is too high when compared to the experiment. It has been found that when serine was considered as nutrients and $D = 8.3 \times
{10^{-6}} {cm^{2}/s}$, the numerical solution was in good agreement with the Keller-Segel solution. The Keller-Segel model also predicts a rather uniform thickness of the leading edge for all values of $\delta/\mu$. But the experimental line shape is fairly symmetric and the leading edge shows variable thickness. The numerical curves tended to be asymmetric[3].

Using the photon correlation spectrometer, Holz and Chen also took into account the presence of tumbling bacteria in formation and movement of migrating bands. They accomplished this by developing a simple analytical model that accounts for the contribution of the twiddle motion to the correlation function. The correlation function was used to analyze microscopic motility characteristics of the bacteria in the band. The result was the intermediate scattering function, which describes the temporal decay of the auto-correlation function and takes into account both the twiddling and running states of the bacteria. It allows for the average fraction of bacteria that are twiddling at a given time to be extracted from the correlation function:

F_{s}(q,t) = {\beta}F_{1}(q,t) + (1-\beta)F_{2}(q,t)
\end{displaymath} (6)

where $F_{1}$ is the intermediate scattering function of bacteria in the twiddle state, $F_{2}$ is the intermediate scattering function of bacteria in the running state, $q$ is the scattering vector, $t$ is time, and $\beta$ is the parameter giving the fraction of bacteria that are twiddling. With proper measurements and estimates, the model and experimental observation agree. The fraction of twiddling bacteria was $\beta = 0.67$, which is reasonably accurate because through visual observation it is easy to see that the bacteria in the migrating band are twiddling extensively[3].

Holz and Chen furthered the study of migrating chemotactic bands by continuing their testing of the Keller-Segel model and by providing means of determining values for the motility ($\mu$) and chemotactic ($\delta$) coefficients of migrating bacteria. They used a rapid-scanning, light-scattering densitometer that produced extensive measurements of band migration speeds and band profiles. To do this, they extended their analysis of the Keller-Segel model by taking into account $D$, the substrate diffusion, which was previously assumed to be zero. They solved the following equation for the concentration of nutrients, taking into account the non-zero diffusion coefficient:

\frac{d^{2}f}{dx^{2}} + {\gamma}\frac{df}{dx} =
\end{displaymath} (7)

where $Q = \frac{k\mu^{2}}{D{\overline{v}}^{2}}RC_0^{\bar\delta
-1}$, $f =\frac{C}{C_{0}}$, where $C$ is the concentration of the substrate, $\gamma = \frac{\mu}{D}$, and $x =
\frac{\overline{v}}{\mu}\xi$. When equation (7) is solved, comparisons can be made between the values for $\mu$ and $\delta$. Through experimentation, Holz and Chen found that $\mu$ was generally in the range of 1-10 $\times 10^{-7}\ cm^{2}/$s and the ratio $\delta/\mu$ was generally in the range of 1.1-2.5. Keller and Segel, using their model, respectively found values of $\mu$ in the range of 1-18 $\times 10^{-7}\ cm^{2}/$s and value for $\delta$ to be in the range of 1.4-2.7. The range of these two sets of values obtained by Holz/Chen and Keller/Segel may be similar on average, but at given instantaneous times, the values of $\mu$ and $\delta$ obtained by Keller and Segel are significantly different from the same values obtained by Holz and Chen[4]. Holz and Chen's experiments showed that Keller and Segel's solution reasonably describes the migration and density profile of the bands, but a more valid analysis is made when taking into account the diffusion of the substrate to obtain values for $\mu$ and $\delta$.

In conclusion, the movement of travelling bands of chemotactic bacteria has been successfully discussed and analyzed. Movement of bacteria is influenced by chemotaxis, a process in which bacteria seek and move toward regions of higher substrate concentration. This movement produces highly visible bands of travelling bacteria. Evelyn Keller and Lee Segel wrote a model to describe this movement by using partial differential equations to describe the consumption of the substrate and the change in bacterial density due to random motion and chemotaxis. They tested their model experimentally, and obtained reasonably accurate measurements for the speed and shape of the travelling bands and also for the ratio of chemotactic strength, $\delta$, to motility, $\mu$. Michael Holz and Sow-Hsin Chen tested Keller and Segel's model and also produced relatively accurate results for the ratio of chemotactic strength to motility and for band shape. They also observed the effects of the twiddling of bacteria in band movement. They correctly predicted that a large number of the bacteria are twiddling, as compared to running in a straight path, at any given time. Additionally, Holz and Chen took into account the diffusion of the substrate in analyzing the Keller-Segel model. In doing this, they obtained values for $\mu$, the motility coefficient, and for $\delta$, the chemotactic coefficient. These values were more accurate than those obtained when neglecting diffusion. Altogether, Keller and Segel reasonably modeled and described the actions of travelling bands of chemotactic bacteria.

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Joceline Lega 2001-10-12