Question

Bacterial foraging Bacteria use swimming to seek out food. Imagine that the bacterium is in a region of low food concentration. For the bacterium to profit from swimming to a region with more food, it has to reach there before diffusion of food molecules makes the concentrations in the two regions the same. Here we find the smallest distance that a bacterium needs to swim so it can outrun diffusion. (a) Make a plot in which you sketch the distance traveled by a bacterium swimming at a constant velocity \(v\) as a function of time \(t\), and the distance over which a food molecule will diffuse in that same time, Indicate on the plot the smallest time and the smallest distance that the bacterium needs to swim to outrun diffusion. (b) Make a numerical estimate for these minimum times and distances for an \(E\). colt swimming at a speed of 30 um/s. The diffusion constant of a rypical food molecule is roughly \(500 \mu \mathrm{m}^{2} / \mathrm{s}\) (c) Estimate the number of ATP molecules the bacterium must consume (hydrolyze) per second in order to travel at this speed, assuming that all of the energy usage goes into overcoming fluid drag. The amount of energy released from one ATP molecule is approximately \(20 \mathrm{k}_{8} \mathrm{T}\). Note that the bacterial flagellar motor is actually powered by a proton gradient and this estimate focuses on the ATP equivalents associated with overcoming fluid drag.

Short answer

The graphical representation will display points where the bacterium and the food molecule lines intersect marking the minimum time and distance to outrun diffusion. The numerical estimation for minimum time and distance can be calculated using the given diffusion constant and velocity. Finally, the number of ATP molecules used per second can be estimated by calculating the energy required to swim.

Step-by-step solution

Sketching the Graph

The graph will have time \( t \) on the horizontal axis and distance \( d \) on the vertical axis. There will be two plots: one for the bacterium, that will be a straight line where the distance traveled by the bacterium is \( vt \), and the other for the food molecule, that will be a curve representing the distance diffused by the food molecule, \( \sqrt{Dt} \). The bacterium would need to swim to the point on the graph where these two lines intersect, marking the minimum distance and time for the bacterium to outrun diffusion.

Numerical Estimates for Time and Distance

To outrun diffusion, the bacterium's distance must be equal to the diffused distance. Hence, equating \( vt \) and \( \sqrt{Dt} \), the bacterium's time can be calculated as \( t = (D/v^2) \). Substituting the given values of \( v = 30 \mu m/s \) and \( D = 500 \mu m^2/s \), calculate \( t \). This calculated time can be substituted into \( vt \) to find the minimum distance.

Energy Estimation

The energy required to swim the minimum distance is given by \( E = FD \), where \( F \) is the fluid drag and \( D \) is the distance traveled. The fluid drag \( F = 6\pi\mu Rv \), where \( R \) is the radius of the bacterium and \( \mu \) is the viscosity of water (use the standard values). Substitute \( F \) in the energy equation to get the energy in terms of ATP molecules by equating it to \( 20k_BT \), where \( k_B \) is the Boltzmann constant and \( T \) is the temperature (use the standard values). Divide this energy by the energy produced per second to get the number of ATP molecules used per second.

Key concepts

Diffusion in Biological Systems

Diffusion is a fundamental process in biological systems, enabling molecules to move from regions of high concentration to areas of lower concentration. In the context of bacterial foraging, diffusion determines how quickly food molecules spread through a given medium. Bacteria aim to exploit this process by reaching higher concentration regions faster than diffusion would equalize the concentrations. To fully understand how bacteria manage this task, we focus on two primary factors:

• The diffusion constant of molecules, often denoted by the symbol \(D\), influences the speed at which a molecule can travel through a fluid medium.
• Bacterium's swimming speed \(v\) determines its ability to reach distant food sources before diffusion balances concentrations.

As food molecules diffuse, they typically spread over a distance approximately proportional to the square root of time multiplied by the diffusion constant, \(\sqrt{Dt}\). By comparing this spread with the linear distance a bacterium can travel, \(vt\), one can identify the precise time and distance where the bacterium outpaces diffusion.

Swimming Behavior of Bacteria

Bacteria exhibit fascinating swimming behavior thanks to structures known as flagella—tail-like appendages that propel them through fluid environments. This motility is essential for bacteria to navigate towards nutrient-rich regions and escape harmful substances. The process can be broadly broken down into two actions:

• Run: A bacterium moves in a straight line powered by the rotation of its flagella in a coordinated manner.
• Tumble: The bacterium changes direction when the flagella rotate in an untimely fashion, causing a random, reorienting motion.

During foraging, bacteria rely on chemotaxis—movement towards or away from chemical stimuli. This behavior ensures they spend more time heading towards desirable environs, optimizing their search efficiency. Their swimming mechanism allows them to cover distances that, if done swiftly enough, outpace environmental diffusion, giving them an evolutionary advantage.

Energy Consumption in Bacteria

Energy consumption is a critical aspect of bacterial swimming. Bacteria must efficiently use their energy reserves while moving, as excessive energy use could deplete resources needed for other vital functions. The energy required to power the bacteria's movement through liquids is primarily used to overcome fluid drag—the resistance of the medium. Bacteria derive their energy from the hydrolysis of ATP molecules, although the flagellar motor is typically powered by proton gradients. For analysis, we often convert this into ATP equivalents. Consider the following:

• The resistance experienced is calculated using the equation for fluid drag, depending on the medium's viscosity and bacteria size.
• The energy per second used ties directly to speed, calculated in terms of how many ATP equivalents are consumed to sustain movement against drag forces.

This calculation allows scientists to understand the efficiency of bacterial motility and predict how changes in environmental factors might impact their foraging behavior.

Bacterial Motility

Bacterial motility is a complex and adaptive trait that allows bacteria to effectively explore their environment and locate resources. Mobilization is critical for survival, aiding in both nutrient acquisition and avoidance of hostile conditions. Motility is largely dependent on the physical and chemical characteristics of the environment and the bacteria's physiological properties. Bacteria move using various methods, but flagella-driven swimming is the most common, which involves rotating appendages to create propulsion. The strategies employed include:

• Chemotaxis: Directed movement in response to chemical gradients, essential for locating nutrients.
• Phototaxis: Movement in response to light stimuli, enabling them to maximize energy acquisition via photosynthesis or evade harmful radiation.

These methods demonstrate the sophistication of bacterial responses to environmental challenges, fuelled by evolutionary adaptations. Studying these aspects not only provides insights into bacterial ecology but also informs biotechnological applications, such as bioremediation and synthetic biology.