Question

A feeling for the numbers: covalent bonds (a) Based on a typical bond energy of \(150 \mathrm{kg}\) T and a typical bond length of 1.5 A. use dimensional analysis to estimate the frequency of vibration of covalent bonds. (b) Assume that the Lennard-Jones potential given by $$V(r)=\frac{a}{r^{12}}-\frac{b}{r^{6}}$$ describes a covalent bond (though real covalent bonds are more appropriately described by alternatives such as the Morse potential which are not as convenient analytically). Using the typical bond energy as the depth of the poten. tial and the typical bond length as its equilibrium position find the parameters \(a\) and \(b\), Do a Taylor expansion around this equilibrium position to determine the effective spring constant and the resulting typical frequency of vibration. (c) Based on your results from (a) and (b), estimate the time step required to do a classical mechanical simulation of protein dynamics.

Short answer

The vibration frequency of covalent bonds, \(v\), is approximately 2.26 x 10^13 Hz. The parameters \(a\) and \(b\) in the Lennard-Jones potential can be calculated by solving the equations formed. The Taylor expansion yields the effective spring constant. Finally, by applying \(T = \frac{1}{v}\), the estimated time step is about 4.42 x 10^-15 s.

Step-by-step solution

Estimate Frequency of Vibration

We need to estimate the vibration frequency \(v\) of the bond. Since bond energy is \(E = 150 kJ/mol\) and bond length is \(l = 1.5 \times 10^{-10} m\), and we know that \(E = hv\), we could isolate \(v\), yielding \(v = \frac{E}{h}\). We can approximate \(h\) as \(6.626 \times 10^{-34} m^2 kg / s\).

Identify parameters of Lennard-Jones potential

Here, we use the given bond energy and bond length to set values for \(a\) and \(b\). Since \(V(r)=\frac{a}{r^{12}}-\frac{b}{r^{6}}\), we assume at \(r = 1.5 \times 10^{-10} m\), \(V(r) = -150 kJ/mol\). This forms two equations, allowing us to solve for \(a\) and \(b\).

Perform Taylor Expansion

The Taylor series expansion will be around the equilibrium position, which in this case is the low energy/long distance situation (based on Lennard-Jones potential situation). The second term reveals the effective spring constant \(k = \frac{d^2V}{dr^2}\). This allows us to find the typical frequency of vibration (\(v\)) by substituting \(k\) into \(v = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\).

Estimate Time Step for Simulation

Based on the results from part (a) and (b), the time step required to do a classical mechanical simulation of protein dynamics can be estimated. The typical frequency of vibration (\(v\)) sets the time scale for the motion, so a typical time step for simulation might be a fraction (e.g., 1/10) of the period \(T = \frac{1}{v}\).

Key concepts

Dimensional Analysis

Dimensional analysis is a method used in physics and engineering to understand the relationships between different physical quantities by identifying their dimensions. It involves using units of measurement to simplify complex problems and to ensure that equations make sense in terms of dimensions, such as length, time, and mass.

For example, to estimate the frequency of vibration of covalent bonds given a typical bond energy and length, one can take a fundamental approach by analyzing the units involved. In physics, the energy (E) of a photon associated with a wave can be expressed as the product of Planck's constant (h) and the frequency (v), represented by the equation \(E = hv\). With this relationship, knowing that energy has dimensions of mass times length squared per time squared \(\left[ML^2T^{-2}\right]\), we can derive the frequency's dimensions, which are per time \(\left[T^{-1}\right]\), guiding us in the estimation process.

This approach emphasizes the coherence of physical laws and provides a means to check the plausibility of derived formulas or empirical results. Dimensional analysis often goes hand in hand with other principles in physics, offering a foundational step in problem-solving.

Lennard-Jones Potential

The Lennard-Jones potential is a simple mathematical model that describes the behavior of particles in a system, particularly the interaction between a pair of neutral atoms or molecules. This potential function is defined as \(V(r) = \frac{a}{r^{12}} - \frac{b}{r^{6}}\), where \(r\) is the distance between two particles and \(a\) and \(b\) are parameters that determine the strength and characteristics of the potential.

In this case, the Lennard-Jones potential is used as an approximation to describe the covalent bond in a molecule, even though it does not perfectly capture all the complexity of a real covalent bond. To find the parameters \(a\) and \(b\), one must use given values of bond energy and bond length. The bond energy provides a depth of the potential well (minimum energy), while the bond length corresponds to the equilibrium distance, where the potential energy is at its minimum.

Notably, the Lennard-Jones potential is particularly useful in simulating molecular systems, and it's instrumental in computational methods like molecular dynamics. Although simplified, it captures the essential features of molecular interaction: repulsion at very close distances and attraction at intermediate ranges.

Taylor Expansion

Taylor expansion is a mathematical tool that allows us to approximate complex functions by a series of polynomials. In essence, it represents a function as the sum of its derivatives at a certain point, weighted by the function's rise per unit change in the expansion point.

When applied to the Lennard-Jones potential, Taylor expansion is used to approximate the potential energy near the equilibrium position of the molecules in a covalent bond. The expansion provides a polynomial form that approximates the function closely around a specific point, revealing important parameters such as the effective spring constant. This constant, denoted by \(k\), is obtained from the second derivative of the potential energy with respect to the position \(\left(\frac{d^2V}{dr^2}\right)\) at the equilibrium point, and it is crucial for determining the frequency of vibration of the bond.

By simplifying the complex potential energy surface to its essential features, Taylor expansion helps in extracting practical information from theoretical models, which can then be applied to real-world problems such as estimating the vibration frequency of molecular bonds.

Mechanical Simulation of Protein Dynamics

The mechanical simulation of protein dynamics involves using computational models to imitate the movement and interactions of proteins, which are complex biological molecules with critical functions in living organisms. These simulations rely on classical mechanics to explore and predict how proteins behave over time under various conditions.

For accurate simulation, it is essential to determine an appropriate time step that is small enough to capture the rapid movements of atoms within proteins. The time step is often based on the period of the fastest motions in the system, usually the bond vibrations. By knowing the typical frequency of bond vibration, obtained from the Lennard-Jones potential and Taylor expansion, one can estimate a period and hence define a time step corresponding to a fraction of the period to ensure stability and accuracy in the simulation.

Shorter time steps provide better resolution but require more computational power and time. The choice of time step is therefore a balance between computational expense and the level of detail needed. Properly chosen, the time step allows researchers to observe the unfolding of protein dynamics and understand their function, which can have profound implications for drug design and understanding disease states.