Problem 86 - Calculus for Biology and Medicine

Question

To measure the physical properties of cells, a piezoelectric probe is used. The force applied by the probe is compared against how much the cell deforms. If \(F\) is the force applied by the probe, and \(w\) is the distance it moves into the cell, then the stiffness of the cell can be calculated from the rate of change, \(d F / d w .\) Zhang et al. found that if \(F\) is measured in \(\mu \mathrm{N}\) and \(w\) in \(\mu \mathrm{m}\) then for a zebrafish embryo: $$ F=3 \times 10^{-4} w^{3}-4.4 \times 10^{-3} w^{2}+3.93 w+0.221 $$ (a) Calculate \(d F / d w\) for this sample. (b) Stiffer cells have larger values of \(d F / d w\) when \(w=0\). Later in embryo development Zhang et al. measure: $$ F=6 \times 10^{-4} w^{3}-5.04 \times 10^{-2} w^{2}+4.08 w+1.12 $$ By calculating \(\frac{d F}{d w} \int_{w-0}\), determine whether the embryo has become stiffer or softer.

Step-by-Step Solution

Answer

Increased stiffness; the embryo is stiffer later as \( dF/dw \) at \( w=0 \) increased from 3.93 to 4.08.

Step 1: Understand the Force Function

The force applied by the probe is given by the function \[ F = 3 \times 10^{-4} w^{3} - 4.4 \times 10^{-3} w^{2} + 3.93w + 0.221. \]Our task in part (a) is to find the derivative of this function with respect to \(w\), which is \(\frac{dF}{dw}\).

Step 2: Differentiate the Force Function for Part (a)

To find \(\frac{dF}{dw}\), we need to take the derivative of each term in the force function separately. - The derivative of \(3 \times 10^{-4} w^{3}\) is \(3 \times 10^{-4} \times 3 w^{2} = 9 \times 10^{-4} w^{2}\).- The derivative of \(-4.4 \times 10^{-3} w^{2}\) is \(-4.4 \times 10^{-3} \times 2 w = -8.8 \times 10^{-3} w\).- The derivative of \(3.93w\) is \(3.93\).- The derivative of the constant \(0.221\) is \(0\).Thus, the resulting derivative is: \[ \frac{dF}{dw} = 9 \times 10^{-4} w^{2} - 8.8 \times 10^{-3} w + 3.93. \]

Step 3: Understand the Second Force Function for Later Development

Later in embryo development, the force is represented by a different function: \[ F = 6 \times 10^{-4} w^{3} - 5.04 \times 10^{-2} w^{2} + 4.08w + 1.12. \]The task for part (b) is to calculate \(\frac{dF}{dw}\) at \(w = 0\) using this new function.

Step 4: Differentiate the Second Force Function

To find \(\frac{dF}{dw}\) for the new force function:- The derivative of \(6 \times 10^{-4} w^{3}\) is \(6 \times 10^{-4} \times 3 w^{2} = 18 \times 10^{-4} w^{2}\).- The derivative of \(-5.04 \times 10^{-2} w^{2}\) is \(-5.04 \times 10^{-2} \times 2 w = -1.008 \times 10^{-1} w\).- The derivative of \(4.08w\) is \(4.08\).- The derivative of the constant \(1.12\) is \(0\).Thus, the resulting derivative is:\[ \frac{dF}{dw} = 18 \times 10^{-4} w^{2} - 1.008 \times 10^{-1} w + 4.08. \]

Step 5: Compare Stiffness at w = 0

Evaluate \(\frac{dF}{dw}\) at \(w = 0\) for both functions:- Initially, \(\frac{dF}{dw} = 3.93\) when \(w = 0\).- Later, \(\frac{dF}{dw} = 4.08\) when \(w = 0\).Since \(4.08 > 3.93\), the embryo becomes stiffer as the derivative increases from the initial to the later stage.

Key Concepts

Derivatives

In calculus, derivatives represent how a function changes as the input changes. They are crucial in understanding how biological systems respond to different conditions. Here, the derivative of the force function with respect to the distance, denoted as \( \frac{dF}{dw} \), tells us how the force changes as the probe moves into the cell. This change is the cell's stiffness.

• To find the derivative, break the function into separate terms and differentiate each individually.
• For \( F = 3 \times 10^{-4} w^{3} - 4.4 \times 10^{-3} w^{2} + 3.93w + 0.221 \), differentiate term by term, treating constants as zero.

Differentiating each term gives:

• \( 3 \times 10^{-4} w^{3} \rightarrow 9 \times 10^{-4} w^{2} \)
• \( -4.4 \times 10^{-3} w^{2} \rightarrow -8.8 \times 10^{-3} w \)
• \( 3.93w \rightarrow 3.93 \)
• Constant \( 0.221 \rightarrow 0 \)

The resulting derivative is: \( \frac{dF}{dw} = 9 \times 10^{-4} w^{2} - 8.8 \times 10^{-3} w + 3.93 \). Here it shows how the force applied by the probe changes with the cell indentation.

Force Function

The force function models the relation between the probe's force and the distance it penetrates the cell. This function helps assess the mechanical properties of cells. Knowing the specific force function is crucial in calculating how stiff or soft an entity is.In the context of our exercise, the initial force function is:\[ F = 3 \times 10^{-4} w^{3} - 4.4 \times 10^{-3} w^{2} + 3.93w + 0.221 \]

• This polynomial equation summarizes how applied force changes as it interacts with the cell.
• Each term of the polynomial considers different powers of \( w \), reflecting how different aspects influence force.

The coefficients and powers of \( w \) in each term show how much each factor contributes to the overall force applied to the cell.

Embryo Development

Embryo development involves many changes, including the mechanical properties of embryonic cells. As embryos develop, their cells often become stiffer. This process is crucial for structural differentiation and the effective creation of tissues and organs.In the exercise:

• The initial stiffness measurement indicates base cell properties at an early stage.
• Later in development, a similar probe measures the force with an updated function:\[ F = 6 \times 10^{-4} w^{3} - 5.04 \times 10^{-2} w^{2} + 4.08w + 1.12 \]
• The derivative at \(w = 0\) increases from 3.93 to 4.08, indicating that cells are stiffer.

Monitoring these changes is vital for understanding growth patterns and structural integrity throughout embryo formation.

Cell Stiffness

Cell stiffness is a measure of how resistant a cell is to deformation when a force is applied. In biological terms, this implies how robust or delicate a cell might be under mechanical stress.How do we quantify stiffness?

• We use the derivative \(\frac{dF}{dw}\) to approximate stiffness. A larger value implies a stiffer cell as more force is needed to achieve the same deformation.
• Stiffer cells resist deformation more than softer ones, ensuring more structural stability.
• Comparing \(\frac{dF}{dw}\) at different developmental stages gives insight into how cells mature or respond to their environment.

In the context of our exercise, changes in cell stiffness between early and later stages of embryo development provide clues on cellular adaptation and structural growth.