Question

In chemotaxis experiments, a source of nutrient molecules can be introduced into the medium containing bacteria via a micropipette. The outward diffusion of the nutrient molecules creates a position-dependent concentration gradient, and the chemotactic response of the bacteria can be observed under a microscope. (a) Estimate the nutrient gradient in steady state as a function of the distance from the micropipette \(r\) by assuming that it keeps the concentration fixed at \(c_{0}\) for distances \(r

Short answer

This exercises demonstrates how the counting error for \(N\) molecules influences the bacterium's ability to detect a nutrient gradient. While the results are dependent on various biological and physical factors, computations illustrate that a bacterium is able to detect a nutrient gradient by maintaining concentration at \(c_{0}\) for distances \(r<r_{0}\). The maximum distance from nutrient source that a bacterium can detect gradient varies with its strategy on receptor arrangement.

Step-by-step solution

Identifying concentration gradient

In steady state, the concentration gradient can be assumed to follow Fick's first law of diffusion which states that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. Hence, we have \(-D \frac{dc}{dr} = J\), where \(J = c_0/4πDr\) (the negative sign indicates direction of flux is opposite to the increase in \(r\)). Therefore, we have \(\frac{dc}{dr} = -\frac{c_0}{4πDr}\) as the equation for concentration gradient where \(D\) is the diffusivity of the nutrient.

Plotting the concentration gradient

To visualize the nutrient gradient, we can plot the values using the equation of the gradient. For this, we will set \(D=1\) (value of \(D\) would not affect the nature of the graph). As a result, concentration gradient is inversely proportional to \(r\), hence it will be highest at \(r=r_0\) and will decrease as \(r\) increases

Maximum distance in bacterium

For the bacterium to detect the gradient, the difference in distances related to the gradient difference must be larger than the measurement error itself. This implies the difference between two measurements is at least \(\sqrt{N} = \sqrt{c_0 a^3}\). Solving for \(r\) in \(c_0 - D \frac{dc}{dr} \cdot 2a > \sqrt{c_0 a^3}\) gives us the maximum detection distance.

Different Receptor Strategy

Now if we consider a different strategy with one receptor, the bacterium compares concentrations at two different positions along a run separated by distance. The error term will be \(2\sqrt{N}\), then we solve for \(r\) in the inequality \(c_0 - D \frac{dc}{dr} \cdot l > 2 \sqrt{c_0 a^3}\) where \(l\) is the distance between two positions.

Key concepts

Nutrient Gradient

A nutrient gradient is a variation in the concentration of nutrients within a medium. In chemotaxis experiments, a micropipette introduces a nutrient source creating such a gradient. The concentration of nutrients decreases with distance from the source. Visualize this as layers of decreasing nutrient concentration moving outward. This gradient guides bacterial movement towards higher nutrient concentrations.
Understanding nutrient gradients is crucial in explaining how bacteria locate and move toward food sources. Chemotaxis, driven by this gradient, is vital for bacteria to sustain themselves, find nutrients, and evade unfavorable conditions.
In a typical chemotaxis experiment, as described, a stable nutrient gradient can be observed under a microscope using specific assumptions and calculations. The nutrient concentration at a position depends on its distance from the micropipette (denoted as "r" in the problem). The interplay between nutrient source and surrounding environment creates a gradient affecting bacterial behavior.

Fick's Law of Diffusion

Fick's Law of Diffusion is a fundamental principle for understanding diffusion processes. It states that the diffusion flux is proportional to the negative of the concentration gradient. Essentially, substances flow from areas of high concentration to those of low concentration.
The general form of Fick's first law as used in this chemotaxis problem is: \[ J = -D \frac{dc}{dr} \]where

• \( J \) is the flux.
• \( D \) is the diffusivity of molecules in the medium.
• \( \frac{dc}{dr} \) is the concentration gradient.

In this case, the equation simplifies the relationship between nutrient concentration and distance, providing a mathematical model for predicting how nutrients spread from the source. This model helps visualize how bacteria sense and respond to changes in nutrient availability over distances.

Bacterial Sensing

Bacteria are remarkable organisms capable of detecting tiny changes in nutrient concentrations to direct their movement towards food sources. This ability to sense and respond is better known as bacterial sensing.
In the context of our problem, the bacterium can detect concentration differences by comparing readings from receptor proteins located at different parts of its body.
The sensitivity of the bacterium to detect these differences is limited by its own physical size and the precision of its biological sensors. The exercise calculations illustrate how a bacterium determines whether the nutrient gradient is strong enough to be sensed. The difference in concentrations across its body must surpass a certain threshold, dictated by the measurement error, for effective movement direction.

Concentration Measurement Error

In experimental terms, measurement error refers to the inconsistency or deviation present when attempting to quantify a particular variable, such as nutrient concentration. For bacteria, detecting nutrient gradients is critical but inherently filled with potential for error. To interpret concentration measurements accurately, bacteria must overcome or minimize errors in their detection systems. The concept described in the problem revolves around counting errors, analyzed using the formula: \[ \text{Error} \approx \sqrt{N} \]where \( N \) is the number of detected molecules.
The exercise emphasizes that the bacterial capacity to detect gradients fundamentally depends on the measurement error being smaller than the actual difference in concentration.
Understanding this aspect helps comprehend challenges faced in biochemical sensing processes and highlights the sophistication of bacterial navigation mechanisms.