Question

There is strong evidence that chemoreceptors in \(E\). coli tend to cluster near one pole (see Kentner and Sourjtk, 2006)\(.\) One hypothesis about the role of such clustering ts that it might increase the ability of a bacterium to better detect molecules in its environment. Determine if this is the most efficient strategy for counting (absorbing) molecules of chemoattractant. Approximate \(E\), coll as a sphere \(a=1 \mu \mathrm{m}\) in radius and neglect its motion. Then compare the diffusive current to \(N=1000\) receptors (absorbing patches of radius \(s=10\) A) scattered over the surface of the cell with the diffusive current to the same receptors incorporated into a single patch with the same total area. Make use of the result that the diffusive cur. rent onto a sphere of radius \(a\) with \(N\) absorbing patches of radius \(s\) spread uniformly over its surface is $$I=\frac{4 \pi D c_{x}}{1+\pi a / N s}$$ where \(D\) is the diffusion constant of the molecules, while \(c_{\infty}\) is their concentration far from the cell. (Adapted from a problem courtesy of H. C. Berg.)

Short answer

The diffusive current for the scattered and clustered receptor configurations are calculated using the given formula. The diffusive currents computed will directly tell us the more efficient strategy for the bacterium to absorb molecules. Whether clustering or scattering is a more efficient strategy can be determined solely by comparing these computed values.

Step-by-step solution

Understand the provided formula

The first step in solving this problem is to understand the provided formula for the diffusive current onto a sphere with \(N\) absorbing patches. The formula is given by \(I=\frac{4 \pi D c_{\infty}}{1+\pi a / N s}\), where \(D\) is the diffusion constant, \(c_{\infty}\) is the concentration far from the cell, \(a\) is the radius of the cell, \(N\) is the number of absorbing patches and \(s\) is the radius of the absorbing patches.

Calculating diffusive current for scattered receptors

The diffusive current when the receptors are scattered over the surface can be calculated using the given formula. Here the radius of the cell \(a = 1 \mu m\), number of receptors \(N = 1000\), and radius of each receptor \(s = 10\) A = \(10^{-8} cm\). Substituting these values in the formula \(I = \frac{4 \pi D c_{\infty}}{1 + \pi a / N s}\) gives us the diffusive current for scattered receptors.

Calculating diffusive current for clustered receptors

In order to calculate the diffusive current for a single large receptor with the same total area, first calculate the total area of all receptors and equate it to the area of the single receptor to find its radius. Since total area is constant, \(N\pi s^2 = \pi S^2\). Here \(s\) and \(S\) are the radii of the single receptors and the single large receptor, respectively. From this, calculate the radius \(S\) of the single large receptor, then calculate the diffusive current using the same formula by replacing \(s\) with \(S\) and \(N\) with 1.

Compare the diffusive currents

Having calculated the diffusive current for scattered receptors and clustered receptors separately, the final step is to compare these values. If the diffusive current for the clustered (single patch) configuration is higher, it would mean that clustering is a more efficient strategy for the bacteria to detect molecules. Otherwise, scattering would be more efficient.

Key concepts

Chemoreceptors in E. coli

The bacteria E. coli possess specialized proteins known as chemoreceptors that are critical for their survival. These chemoreceptors enable E. coli to detect changes in the chemical composition of their environment, specifically the presence of chemoattractants or repellents. When a chemoattractant molecule binds to a chemoreceptor, it triggers a signal transduction pathway that influences the bacterium's swimming behavior, guiding it towards favorable environments.

The interesting clustering of these chemoreceptors at one pole of the E. coli cell is hypothesized to enhance the bacterium's ability to detect molecules more efficiently. The rationale is that a denser array of receptors may increase the probability that chemoattractants will bind to at least one receptor, thereby initiating a signaling response more readily. This spatial organization could significantly impact the molecular detection efficiency, as it would reduce the response time and might enhance sensitivity to weaker signals.

Molecular Detection Efficiency

Molecular detection efficiency can be thought of as the ability of a cell's receptors to capture and respond to external chemical molecules effectively. This is an essential function in cellular signaling and behavior. Several factors play a role in detection efficiency, including the number of receptors, their spatial distribution on the cell surface, and the binding affinity of receptors to the molecules.

Detection efficiency is further influenced by factors such as receptor clustering, which can potentially enhance sensitivity by creating 'hot spots' for molecule capture. For instance, in the case of E. coli, if chemoreceptors are clustered together, they may provide a higher local concentration of detection sites, possibly increasing the likelihood of molecular capture due to the reduced path a molecule must diffuse to encounter a receptor.

Diffusion Constant

The diffusion constant, denoted by D, is a crucial parameter that describes how fast molecules move in a medium due to diffusion - a process driven by random thermal motion. This constant is determined by factors such as temperature, viscosity of the medium, and size of the diffusing molecules.

Understanding the diffusion constant is vital in assessing how quickly molecules can reach the cell's receptors. In our exercise, the diffusive current formula includes the diffusion constant, which, along with the concentration gradient of the molecules and the geometry of the receptors, determines how efficiently a molecule can be captured by the cell. The formula given, I = \(\frac{4 \(pi\) D c_{\(infty\)}}{1 + \(pi\) a / N s}\), illustrates how the geometry and number of receptors interact with the diffusion constant to influence the ultimate rate of molecular detection by the cell. In simpler terms, a higher diffusion constant means that molecules move faster and are more likely to come into contact with the cell's receptors.