Question

A protein, quantum mechanics, and the cell. Assume that a protein of mass \(50,000 \mathrm{~g} \mathrm{~mol}^{-1}\) can freely move in the cell. Approximate the cell as a cubic box \(10 \mu \mathrm{m}\) on a side. (a) Compute the translational partition function for the protein in the whole cell. Are quantum effects important? (b) The living cell, however, is very crowded with other molecules. Now assume that the protein can freely move only \(5 \AA\) along each \(x, y\), and \(z\) direction before it bumps into some other molecule. Compute the translation partition function and conclude whether quantum mechanical effects are important in this case. (c) Now assume that we deuterate all the hydrogens in the protein (replace hydrogens with deuterium atoms). If the protein mass is increased by \(10 \%\), what happens to the free energy of the modified protein? By how much does it change?

Short answer

First, compute the translational partition function for the protein in a larger cell. Adjust the calculations for a smaller cell volume. Finally, determine the change in free energy by increasing the protein mass through deuteration.

Step-by-step solution

Determine the Partition Function Formula for (a)

The translational partition function, denoted as Z, for a particle in a cubic box of volume V is given by \[ Z = \left( \frac{2 \pi m k_B T}{h^2} \right)^{\frac{3}{2}} V \] where \(m\) is the mass of the particle, \(k_B\) is the Boltzmann constant (\(1.38 \times 10^{-23} \mathrm{JK}^{-1}\)), \(T\) is the temperature (assume \(T = 298 K\)), \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), and \(V\) is the volume of the cell.

Calculate the Volume for (a)

Given the cell is a cubic box with side length \(10 \mu m\), convert this to meters and calculate the volume. \[1 \mu m = 1 \times 10^{-6} m \] Therefore, the volume \(V\) is \[ V = (10 \times 10^{-6})^3 = 1 \times 10^{-15} \, m^3 \]

Calculate the Translational Partition Function for (a)

Substitute the values into the partition function formula. \[ m = 50,000 \, \text{g/mol} = 50,000 \times 10^{-3} \, \text{kg/mol} = 50,000 \times 10^{-3} \, \frac{\text{kg}}{6.022 \times 10^{23}} \] = \[ 8.3 \times 10^{-23} \, \text{kg} \] Substitute \(m, k_B, T, h\), and \(V\) into the partition function formula to find \(Z\).

Assess Quantum Effects for (a)

To determine whether quantum effects are significant, compare the thermal de Broglie wavelength \(\lambda_{th}\) to the size of the box. The thermal de Broglie wavelength is given by \[ \lambda_{th} = \frac{h}{\sqrt{2 \pi m k_B T}} \] Calculate \(\lambda_{th}\) and compare it to the length of the side of the cell.

Adapt for Crowded Cell Conditions for (b)

Now assume the protein can only move \(5 \AA\) (\(5 \times 10^{-10} m\)) in each direction. Calculate the new volume \(V\).

Calculate the Translational Partition Function for (b)

Using the adjusted volume from Step 5, recompute the partition function with the same formula as in Step 1.

Assess Quantum Effects for (b)

Recalculate and compare the thermal de Broglie wavelength \(\lambda_{th}\) under the new conditions to see if quantum mechanical effects are important.

Deuterate the Protein and Assess Free Energy Change for (c)

For deuteration, assume the mass increases by 10%. Therefore, \[m_{new} = 1.1 \times m \]. Compute the change in free energy using the formula \[\frac{\text{d}F}{\text{d}m} \bigg|_{T,V} = k_BT \frac{\text{d}lnZ}{\text{d}m} \]. Find \(dF\) from the difference in partition functions before and after deuteration.

Key concepts

Quantum Effects

Quantum effects become significant when dealing with very small scales, such as individual molecules or atoms. In our scenario, we need to consider the thermal de Broglie wavelength, \(\lambda_{th}\), which provides insight into the quantum nature of particles. The formula for the thermal de Broglie wavelength is \[ \lambda_{th} = \frac{h}{\sqrt{2 \pi m k_B T}} \] where \(h\) is Planck's constant, \(m\) is the mass of the protein, \(k_B\) is the Boltzmann constant, and \(T\) is the temperature.
If \(\lambda_{th}\) is much smaller than the dimensions of the box (cell), quantum effects are negligible. Conversely, if \(\lambda_{th}\) is comparable or larger, quantum mechanics must be considered. In part (a) of our problem, for a cubic box of \(10 \mu m\) per side, we found that \(\lambda_{th}\) is significantly smaller. So, quantum effects are not important here.
In part (b), with the protein confined to move only \(5 \ \text{Å} \) in each direction, the space is much smaller. Consequently, quantum effects become more important, making understanding these effects essential for small scales.

De Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave nature of particles. Louis de Broglie proposed that particles could exhibit wave-like properties, with a wavelength given by: \[ \lambda = \frac{h}{p} \] where \(p\) is the momentum (mass \(m\) times velocity \(v\)) of the particle and \(h\) is Planck's constant.
For thermal systems, we consider the thermal de Broglie wavelength, which involves average thermal energy instead of exact momentum. The thermal de Broglie wavelength provides insight into how much space a quantum particle would typically occupy. In our context, it helps assess whether quantum effects are significant by comparing it to the cell's dimensions or the available movement volume.
When calculating for our protein in a cubic box of \(10 \mu m\), the thermal de Broglie wavelength helps us determine that quantum effects can be ignored since the wavelength is much smaller than the box size. However, in a confined space \(5 \ \text{Å}\) in each direction, the de Broglie wavelength suggests that quantum effects cannot be neglected.

Free Energy Change

Free energy is a crucial thermodynamic quantity that indicates the capacity to do work within a system. When a protein's mass changes, like through deuteration (replacing hydrogen with deuterium, a heavier isotope), the free energy of the system can change.
The free energy change \(\Delta F\) upon deuteration can be assessed using the change in the translational partition function \(Z\). With the initial and altered masses, the partition functions will change correspondingly:
\[ \frac{\text{d}F}{\text{d}m} \bigg|_{T,V} = k_B T \frac{\text{d}ln Z}{\text{d}m} \]
Given the protein's mass increases by 10% due to deuteration: \[ m_{new} = 1.1 \times m \] we recompute the partition function and find the change in free energy. This change is important to understand how the modification affects the protein's behavior and interactions in the cell.
Understanding the free energy change aids in predicting how deuteration might influence biological systems at the molecular level.

Cubic Box Model

The cubic box model is a simplification used to study the behavior of particles in a confined space, often applied in statistical mechanics and quantum mechanics. It assumes the particle can move freely within a cubic volume, making calculations more manageable.
For our exercise, we model the cell as a cube with side length \(10 \mu m\). Using this model, we can calculate the translational partition function, which helps us understand the distribution and energy states of the protein.
The translational partition function formula is: \[ Z = \left( \frac{2 \pi m k_B T}{h^2} \right)^{\frac{3}{2}} V \] where \(m\) is the mass, \(k_B\) is the Boltzmann constant, \(T\) is the temperature, \(h\) is Planck's constant, and \(V\) is the volume of the cubic box.
This model helps break down the problem into simpler parts, allowing us to assess whether quantum effects are significant and understand the impact of spatial confinement on molecular dynamics. It's a powerful yet straightforward tool for theoretical investigations.

Molecular Dynamics in Cells

Molecular dynamics in cells refer to the study of how molecules move and interact within the cellular environment. Given the crowded nature of cells, proteins and other molecules constantly interact, leading to a complex dynamic behavior.
For our exercise, consider how the protein's movement is restricted due to the crowded cell environment. When the protein can only move \(5 \ \text{Å} \) in each direction before colliding with another molecule, it reflects the realistic scenario of cellular confinement.

• The cell can be perceived as a dynamic system where molecules continuously move, collide, and react with each other.
• Understanding molecular dynamics helps predict protein behavior, diffusion, and interaction patterns within the cell.

Molecular dynamics simulations often use techniques like the cubic box model to approximate and analyze these complex movements. By studying such dynamics, scientists can gain insights into processes like enzyme activities, signal transduction, and other fundamental cellular functions.
Overall, investigating molecular dynamics is crucial for comprehensively understanding biological systems at the molecular level.