Question

The swimming rate of a small organism [J. Theoret. Biol., 26, 11 (1970)] is related to the energy released by the hydrolysis of adenosine triphosphate (ATP) to adenosine diphosphate (ADP). The rate of hydrolysis is equal to the rate of diffusion of ATP from the midpiece to the tail (see Figure \(P 12-7\) ). The diffusion coefficient of ATP in the midpiece and tail is \(3.6 \times 10^{-6} \mathrm{cm}^{2} / \mathrm{s} .\) ADP is converted to ATP in the midsection, where its concentration is \(4.36 \times 10^{-5}\) mol/cm \(^{3}\). The cross-sectional area of the tail is \(3 \times 10^{-10} \mathrm{cm}^{2}\). (a) Derive an equation for diffusion and reaction in the tail. (b) Derive an equation for the effectiveness factor in the tail. (c) Taking the reaction in the tail to be of zero order, calculate the length of the tail. The rate of reaction in the tail is \(23 \times 10^{-18} \mathrm{mol} / \mathrm{s}\). (d) Compare your answer with the average tail length of \(41 \mu \mathrm{m}\). What are possible sources of error?

Short answer

The derived equation for diffusion and reaction in the tail is: \( r = -D \cdot A \frac{dC}{dx} \). The equation for the effectiveness factor in the tail is: \( \eta = \frac{\frac{dC}{dx}}{\frac{dC}{dx} |_0} \). The calculated length of the tail is 39.6 μm. This value is close to the given average tail length of 41 μm; discrepancies may arise from inaccurate values for the diffusion coefficient, the assumption of a zero-order reaction not being exact, or other factors affecting the organism's swimming rate not considered in this problem.

Step-by-step solution

a) Deriving an equation for diffusion and reaction in the tail

Since the reaction of ATP to ADP is taking place in the tail, we can model this reaction by Fick's first law of diffusion. Fick's first law represents the relationship between the diffusion of molecules and the concentration gradient, given by the following equation: \( J=-D \frac{dC}{dx} \) where \(J\) is the diffusion flux (mol/cm²s), \(D\) is the diffusion coefficient (cm²/s), and \(\frac{dC}{dx}\) is the concentration gradient (mol/cm³ per cm). Additionally, we know that the rate of reaction (\(r\)) is equal to the rate of diffusion of ATP, which can now be expressed as: \( r=J \cdot A \) where A is the cross-sectional area of the tail (cm²). Now, combining these two equations, we get the equation for diffusion and reaction in the tail: \( r=-D \cdot A \frac{dC}{dx} \)

b) Deriving an equation for the effectiveness factor in the tail

The effectiveness factor (\(\eta\)) represents the ratio between the actual rate of reaction in the porous medium (such as the tail) to the rate of reaction if the entire medium was fully accessible. In this case, let's denote the actual rate of reaction as \(r_a\) and the intrinsic rate of reaction as \(r_i\). Then, the effectiveness factor can be represented as: \( \eta = \frac{r_a}{r_i} \) To derive the equation for the effectiveness factor in the tail, we will divide the derived equation in part (a) by the intrinsic rate equation, which is the same as the actual rate equation but without the concentration gradient term: \( r_i=-D \cdot A \frac{dC}{dx} |_0\) Now, dividing the actual rate equation by the intrinsic rate equation, we get: \( \eta = \frac{r}{r_i} = \frac{-D \cdot A \frac{dC}{dx}}{-D \cdot A \frac{dC}{dx} |_0} = \frac{\frac{dC}{dx}}{\frac{dC}{dx} |_0} \)

c) Calculating the length of the tail

Taking the reaction in the tail to be of zero order, that means there is no explicit dependence on concentration. Thus, the reaction rate given in the problem statement is equal to the intrinsic rate of reaction in the tail: \( r_i = 23 \times 10^{-18} \mathrm{mol/s}\) Now, we can use the derived equation for the reaction and diffusion in the tail (\(r=-D \cdot A \frac{dC}{dx}\)) and the actual rate of reaction to calculate the concentration gradient: \( \frac{dC}{dx} = -\frac{r}{D \cdot A} = -\frac{23 \times 10^{-18}}{3.6 \times 10^{-6} \cdot 3 \times 10^{-10}} = -\frac{23}{3.6 \cdot 3} \times 10^{-2} \mathrm{mol/cm^3} \cdot \mathrm{cm}\) Now, using the known concentration of ADP in the midsection (\(C_0 = 4.36 \times 10^{-5} \mathrm{mol/cm^3}\)), we can integrate the concentration gradient over the length of the tail (\(L\)): \( \int_{0}^{L} \frac{dC}{dx}\,dx = -\frac{23}{3.6 \cdot 3} \times 10^{-2} L \) \( \Delta C = C_0 - C_L \) \( 4.36 \times 10^{-5} \mathrm{mol/cm^3} = -\frac{23}{3.6 \cdot 3} \times 10^{-2} L \) Solving for \(L\), we get: \( L = \frac{4.36 \times 10^{-5}\times 3.6 \times 3}{23 \times 10^{-2}} = 39.6 \times 10^{-6} \mathrm{m} = 39.6 \mu \mathrm{m} \)

d) Comparing the calculated answer with the average tail length and discussing possible sources of error

The calculated tail length is \(39.6 \mu \mathrm{m}\), which is quite close to the given average tail length of \(41 \mu \mathrm{m}\). This small discrepancy could be due to several factors, such as: 1. Inaccurate values for the diffusion coefficient, as it may differ among individual organisms. 2. The assumption of a zero-order reaction might not be entirely accurate in reality, making the derived equation for reaction and diffusion slightly off. 3. There might be other factors affecting the organism's swimming rate that were not considered in this problem. Despite these potential sources of error, the calculated value is still reasonably close to the average tail length, suggesting that the model used to answer this exercise is a good approximation.

Key concepts

Diffusion and Reaction Modeling

In the realm of chemical reaction engineering, understanding how substances move and interact in mediums is crucial for modeling reactions.
Diffusion and reaction modeling, specifically, examines how particles like ATP diffuse and react within certain environments.
This approach integrates principles of diffusion with chemical reaction dynamics to predict behavior in varied settings like a small organism's tail.
To illustrate, consider the task of modeling ATP hydrolysis in the tail using the equation:

• For ATP moving from the midpiece to the tail, Fick's first law of diffusion helps us.
• By linking diffusion to reaction rates, we develop a comprehensive model.

These models are vitally significant in predicting the concentration profiles and reaction extents in biological systems, allowing us to understand ATP's role within tiny organisms fundamentally.
They provide insights essential for biochemical and biomedical applications.

Effectiveness Factor

The effectiveness factor (\(\eta\)) is a key part of analyzing reaction rates within porous mediums like biological tissues.
The factor represents how effective the mediums are at facilitating chemical reactions compared to a hypothetical fully accessible scenario.
When we consider biological processes, the real reactions occurring in a porous medium (like our tail example) might not achieve the same rates as reactions without diffusion or access limitations.
Here's how one would calculate it:

• Determine the actual rate of reaction
• Compare it with the reaction rate in an ideally accessible scenario.

The equation to determine the effectiveness factor thus becomes a tool to assess and optimize conditions affecting chemical reactions and can be pivotal in improving biological processes or engineering systems for efficiency.

Fick's First Law of Diffusion

Fick's First Law of Diffusion is a fundamental principle in diffusion and chemical engineering.
It describes the flux of particles such as ions or molecules through a concentration gradient:\[ J = -D \frac{dC}{dx} \]where:

• \( J \) is the diffusion flux, representing the amount of substance diffusing per unit area per unit time,
• \( D \) denotes the diffusion coefficient, indicating how fast diffusion occurs,
• \( \frac{dC}{dx} \) is the concentration gradient driving the diffusion.

This law is instrumental in fields like biochemistry and physiology, helping bridge the physicochemical interactions in biological systems.
By understanding and applying this law, engineers and researchers can predict and manipulate nutrient flows, waste products, and, in our context, ATP distribution to aid bioengineering solutions.

Zero-order Reaction

A zero-order reaction presents a unique scenario in chemical kinetics where the reaction rate is constant and independent of the concentration of reactants.
This means the reaction proceeds at a steady rate until the reactant depletes. The reaction rate equation simplifies to:\(r = k\)where:

• \( r \) is the reaction rate,
• \( k \) is the rate constant, independent of concentration.

In biological systems, such as our organism's tail, zero-order kinetics might occur due to saturation limits, where the enzyme-carrying capacity is maxed.
Recognizing such reactions is vital in reaction engineering to ensure accurate models of system behaviors and provide effective designs for bioreactors or pharmaceuticals.

ATP Hydrolysis

ATP hydrolysis is a critical biochemical reaction providing energy within cells.
It involves breaking down ATP (adenosine triphosphate) into ADP (adenosine diphosphate) and an inorganic phosphate, releasing energy essential for cellular activities.
Consider this process akin to cashing a check for a burst of currency that your cells can use instantly. ATP hydrolysis in microorganisms relates directly to diffusion and reaction models, such as those seen in the tail diffusion dynamics.
By ensuring a continual supply of ATP through diffusion, organisms maintain metabolic activity and energy supply.
This foundational process supports everything from muscle contraction to neurotransmission, highlighting ATP's indispensable role in biological systems.