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 concept of morphogen gradient: A morphogen is a substance that defines different cell fates in a concentration-dependent manner. In this problem, the morphogen concentration decreases by half every 100 µm from the posterior pole of the Drosophila embryo.
2. Identify the initial concentration at the posterior pole: Let's denote the initial concentration of the morphogen at the posterior pole as C₀.
3. Determine the concentration required for leg formation: The cell requires 1/16th of the initial concentration (C₀) to form part of a leg. Therefore, the required concentration is C₀/16.
4. Calculate the distance where the concentration is C₀/16: Since the concentration halves every 100 µm, we can express the concentration at a distance x as C₀/(2^(x/100)). Set this equal to C₀/16 and solve for x using the equation: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> </mrow> <mrow> <msup> <mn>2</mn> <mfrac> <mi>x</mi> <mn>100</mn> </mfrac> </msup> </mrow> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>C</mi> <mn>0</mn> </msub> <mn>16</mn> </mfrac> </math>
5. Solve the equation for x: Simplify the equation to find the value of x that satisfies the condition. This involves solving the equation <math xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mn>2</mn> <mfrac> <mi>x</mi> <mn>100</mn> </mfrac> </msup> <mo>=</mo> <mn>16</mn> </math> which can be solved by recognizing that 16 is 2 raised to the power of 4, thus x/100 = 4, leading to x = 400 µm.

Key Concepts

Morphogen Gradient

A morphogen gradient is a concentration gradient of a signaling molecule, known as a morphogen, which helps determine the fate of cells in a developing embryo. Cells interpret different concentrations of morphogens to activate specific developmental pathways. In this scenario, the morphogen concentration decreases by half every 100 µm from the posterior end, creating a gradient that influences cell differentiation.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the context of the morphogen gradient, the concentration halves every 100 µm, indicating an exponential decay pattern. Understanding this concept is crucial for calculating the distance at which the morphogen concentration reaches a specific fraction of its initial value.

Logarithmic Calculation

Logarithmic calculation is used to solve problems involving exponential changes, such as determining the distance at which a morphogen concentration reaches a specific fraction of its initial value. By applying the properties of logarithms, one can calculate the number of halving events needed to reach 1/16th of the initial concentration, which corresponds to the distance from the morphogen source.