Imagine a situation in which a morphogen has its source at the posterior end of a Drosophila embryo. Every 100 µm from the posterior pole, the morphogen concentration decreases by half. If a cell required 1/16th the amount of morphogen found at the posterior pole to form part of a leg, how far from the posterior pole would the leg form? a. 100 μm b. 160 μm c. 400 μm d. 1600 μm

Verified step by step guidance

1. Understand the problem: The morphogen concentration decreases by half every 100 µm from the posterior pole. We need to find the distance where the concentration is 1/16th of the initial concentration at the posterior pole.
2. Recognize the pattern: The concentration halves every 100 µm. This forms a geometric sequence where each term is half of the previous term. The sequence starts at 1 (full concentration) and halves at each step: 1, 1/2, 1/4, 1/8, 1/16, ...
3. Identify the term: We need to find the distance where the concentration is 1/16th. From the sequence, 1/16th is the fifth term.
4. Calculate the distance: Since the concentration halves every 100 µm, and 1/16th is the fifth term, the distance is 100 µm multiplied by 4 (since the first term is at 0 µm).
5. Answer: The leg would form 400 µm from the posterior pole.

Key Concepts

Morphogen Gradients

Morphogens are signaling molecules that dictate cell fate based on their concentration gradients. In this scenario, the morphogen concentration decreases exponentially as one moves away from its source, which is crucial for understanding how cells interpret these signals to determine their developmental roles.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In this case, the morphogen concentration halves every 100 µm, illustrating how quickly the signal diminishes and how this affects the distance at which specific cellular responses, like leg formation, can occur.

Threshold Concentration

The threshold concentration is the minimum amount of a morphogen required for a cell to respond and differentiate into a specific type. Here, the leg-forming cells need 1/16th of the concentration found at the posterior pole, which helps determine the distance from the source where this threshold is met.