Problem 23 - Calculus for Biology and Medicine

Question

Molecular Dynamics One popular model for the interactions between two molecules is the Leonard-Jones \(6-3\) potential. According to this model, the energy of interaction between two molecules that are distance \(r\) apart is given by a function: $$ V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}} r>0 $$ Molecules will attract or repel each other until they reach a distance that minimizes the function \(V(r)\). The coefficient \(A\) is a positive constant. (a) What is the behavior of \(V(r)\) as \(r \rightarrow 0 ?\) What is the behavior of \(V(r)\) as \(r \rightarrow+\infty\) ? (b) Explain why you expect there to be a value of \(r\) that minimizes \(V(r)\), and then calculate that value of \(r\) (it may help for your argument to determine the sign of \(V^{\prime}(r)\) for large \(\left.r\right)\). (c) Would you still expect there to be a spacing that minimizes \(V(r)\) if \(A\) were a negative number? Justify your answer.

Step-by-Step Solution

Answer

(a) As \( r \to 0 \), \( V(r) \to \infty \); as \( r \to \infty \), \( V(r) \to 0 \). (b) \( r = \left( \frac{2}{A} \right)^{1/3} \) minimizes \( V(r) \). (c) If \( A < 0 \), no minimum exists for \( V(r) \).

Step 1: Analyze Behavior as r Approaches Zero

As \( r \to 0 \), the term \( \frac{1}{r^6} \) dominates because its growth rate is much faster than any term divided by \( r^3 \). Therefore, as \( r \to 0 \), \( V(r) \) approaches infinity. This indicates the potential energy grows positively without bound, suggesting a strong repulsion.

Step 2: Analyze Behavior as r Approaches Infinity

As \( r \to \infty \), both terms \( \frac{1}{r^6} \) and \( \frac{A}{r^3} \) approach zero. However, because the \( \frac{A}{r^3} \) term dominates initially (for large but finite \( r \)), \( V(r) \) approaches zero. This suggests that the potential energy vanishes as molecules are infinitely apart.

Step 3: Find Value of r that Minimizes V(r)

To find the minimum of \( V(r) \), we calculate its derivative and set it to zero: \[ V'(r) = -\frac{6}{r^7} + \frac{3A}{r^4} = 0. \]Simplifying gives:\[ \frac{3A}{r^4} = \frac{6}{r^7} \rightarrow 3A = \frac{6}{r^3} \rightarrow r^3 = \frac{2}{A}. \]Thus, \( r = \left( \frac{2}{A} \right)^{1/3} \) is the distance that minimizes \( V(r) \).

Step 4: Determine Effect of Negative A on Minimization

If \( A \) were negative, the term \( -\frac{A}{r^3} \) would instead be positive, causing \( V(r) \) to be dominated by this term for all positive \( r \). In this case, as \( r \rightarrow 0 \), \( V(r) \rightarrow \infty \), and \( V(r) \rightarrow \infty \) as well for large \( r \), indicating no minimum. Thus, no real value of \( r \) would minimize the potential.

Key Concepts

Leonard-Jones Potential

Leonard-Jones potential is a mathematical model, used to describe how two molecules interact at a distance. It establishes a formula for potential energy based on the distance between molecular centers, specifically exploring both the attractive and repulsive forces.

Imagine two molecules that behave like magnets. When they are extremely close, they repel each other forcefully, akin to the sides of magnets that push apart. This is captured when the function rises sharply as the distance gets tiny, due to the term \( \frac{1}{r^6} \). As they move further apart, they experience attraction, like opposite magnetic poles pulling together, dominated by the \( \frac{A}{r^3} \) term.

The Leonard-Jones potential is instrumental in molecular dynamics simulations, providing insight into how molecular structures settle into the most stable form, with minimum potential energy.

Molecular Interactions

Molecular interactions are the physical forces between molecules that can either pull them together or force them apart. These interactions are fundamental in determining the physical properties of substances, such as boiling points and solubility.

• **Attractive Forces**: These include van der Waals forces and other dipole interactions, which stabilize molecules at moderate distances.


• **Repulsive Forces**: When molecules come very close to each other, electron clouds repel, leading to increased potential energy, as explained by the term \( \frac{1}{r^6} \).

Through the lens of the Leonard-Jones potential, we can predict the equilibrium positioning of molecules in substances, leading to a more comprehensive understanding of chemical behavior in different scenarios.

Potential Energy

Potential energy in molecular dynamics is the stored energy based on the position and configuration of molecules. In the context of Leonard-Jones potential, it signifies the energy landscape that dictates how molecules interact within a given space.

As distance \( r \) changes, potential energy shifts. When molecules are at the optimal distance, potential energy is minimized, facilitating stable arrangements. Think of this as molecules finding a comfortable resting position where they neither repel nor attract each other excessively.

• When molecules are too close, potential energy spikes due to repulsive interactions.


• When too far, the attractive forces weaken, raising potential energy again complementary.

Understanding potential energy sheds light on molecular stability, crucial for fields ranging from chemistry to biological systems.

Differential Calculus

Differential calculus is a mathematical tool for analyzing functions and finding rates of change. In the context of the Leonard-Jones potential, it's used to uncover points of minimal potential energy by finding where the derivative of the potential function equals zero.

To determine where molecules are most stable, we calculate the first derivative (\( V'(r) \)) and find its roots, indicating potential minima or maxima.

• Finding \( V'(r) = 0 \) helps locate the minimal energy distance \( r \), where molecules prefer to settle.


• It assists in identifying whether solutions are stable configurations (minima) or unstable (maxima).

With differential calculus, scientists can design simulations more accurately, predicting how molecules will behave under various conditions.