Question

a. The diffusion coefficient of the protein lysozyme \((\mathrm{MW}=14.1 \mathrm{kg} / \mathrm{mol})\) is \(0.104 \times 10^{-5} \mathrm{cm}^{2} \mathrm{s}^{-1} .\) How long will it take this protein to diffuse an rms distance of \(1 \mu \mathrm{m} ?\) Model the diffusion as a three-dimensional process. b. You are about to perform a microscopy experiment in which you will monitor the fluorescence from a single lysozyme molecule. The spatial resolution of the microscope is \(1 \mu \mathrm{m} .\) You intend to monitor the diffusion using a camera that is capable of one image every 60 s. Is the imaging rate of the camera sufficient to detect the diffusion of a single lysozyme protein over a length of \(1 \mu \mathrm{m} ?\) c. Assume that in the microscopy experiment of part (b) you use a thin layer of water such that diffusion is constrained to two dimensions. How long will it take a protein to diffuse an rms distance of \(1 \mu \mathrm{m}\) under these conditions?

Short answer

In summary, the lysozyme protein takes approximately 2.56 seconds to diffuse an rms distance of 1 µm in a 3-dimensional space and 3.85 seconds in a 2-dimensional space. The camera's imaging rate of one image every 60 seconds is not sufficient to detect the protein's diffusion over a length of 1 µm.

Step-by-step solution

a. Time for protein to diffuse an rms distance of 1 µm in 3-dimensional space

Given the diffusion coefficient (D) of the protein is \(0.104 \times 10^{-5} \thinspace cm^2s^{-1}\), we need to determine the time (t) it takes for the protein to diffuse an rms distance of 1 µm (or \(10^{-4}\) cm). We can use the 3-dimensional rms distance formula: \(x_{rms} = \sqrt{6Dt}\) Rearrange the formula for time (t): \(t = \frac{x_{rms}^2}{6D}\) Plug in the given values: \(t = \frac{(10^{-4})^2}{6\times (0.104 \times 10^{-5})}\) Calculate the time: \(t \approx 2.56 \thinspace s\) Hence, it will take approximately 2.56 seconds for the lysozyme protein to diffuse an rms distance of 1 µm in a 3-dimensional space.

b. Determining if the camera's imaging rate is sufficient to detect the protein's diffusion

The camera being used is capable of capturing an image every 60 seconds. We need to determine if this imaging rate is sufficient to detect the diffusion of a single lysozyme protein over a length of 1 µm. From part a, we know that the time required for the protein to diffuse an rms distance of 1 µm is about 2.56 seconds. Since 2.56 seconds is much less than the imaging rate of the camera (60 seconds), the imaging rate is NOT sufficient to detect the diffusion of a single lysozyme protein over a length of 1 µm.

c. Time for protein to diffuse an rms distance of 1 µm in 2-dimensional space

Here, we assume that the diffusion process is constrained to two dimensions. We can use the 2-dimensional rms distance formula: \(x_{rms} = \sqrt{4Dt}\) Rearrange the formula for time (t): \(t = \frac{x_{rms}^2}{4D}\) Plug in the given values: \(t = \frac{(10^{-4})^2}{4\times (0.104 \times 10^{-5})}\) Calculate the time: \(t \approx 3.85 \thinspace s\) Therefore, it will take approximately 3.85 seconds for the lysozyme protein to diffuse an rms distance of 1 µm in a 2-dimensional space.

Key concepts

Diffusion Coefficient

The diffusion coefficient, denoted as 'D', is a vital parameter indicating how fast a particular substance such as a protein can diffuse through a medium. In a biological context, the diffusion coefficient provides insights into the movement of molecules like proteins within the cell or extracellular environment.

For lysozyme, a protein with a molecular weight of 14.1 kg/mol, its diffusion coefficient is given as 0.104 × 10^-5 cm^2/s. This small numerical value might not say much at a glance, but it translates to how quickly the protein molecules spread out from an initial concentration over time. A high value of 'D' signifies rapid diffusion, while a lower value indicates slower molecular movement.

RMS Distance

The root mean square (rms) distance is a measure representing the average distance a diffusing particle, such as a protein, will travel over a certain period. In our case, the rms distance plays a role in expressing how far the protein lysozyme may diffuse within the given frame. For the lysozyme protein to diffuse an rms distance of 1 μm, one would calculate the time using the established formula, incorporating the previously mentioned diffusion coefficient 'D'.

This concept is crucial in understanding how particles move in biological systems and can have implications on various processes such as signaling, metabolism, and drug delivery within the body.

Molecular Diffusion

Molecular diffusion is a process by which molecules spread from regions of high concentration to low concentration as a result of their random motion. It is an essential mechanism for the movement of substances at the microscopic level, including the distribution of proteins within cellular and extracellular fluids. Factors that affect the rate of molecular diffusion include temperature, molecular size, and the viscosity of the surrounding medium.

Molecular diffusion is driven by the kinetic energy within the system, leading to a net movement until equilibrium is achieved. It's particularly significant in biological systems where it facilitates various critical functions, including nutrient uptake and waste removal.

Three-Dimensional Diffusion

In three-dimensional diffusion, molecules move freely in all spatial dimensions—X, Y, and Z axes. This process can be imagined like the distribution of a scent in a room where the molecules disperse in all available directions. This model is frequently used to describe the diffusion within bulk solutions, such as a protein like lysozyme in the intracellular fluid.

For the protein to traverse an rms distance of 1 μm in a three-dimensional space, our calculation estimates a time of approximately 2.56 seconds. However, this is an averaged prediction and actual rates can depend on various in situ factors. The concept of three-dimensional diffusion is a cornerstone in understanding how molecular interactions occur within living organisms where cells and organelles operate in a complex 3D environment.

Two-Dimensional Diffusion

Contrary to the three-dimensional counterpart, two-dimensional diffusion occurs when the movement of molecules is limited to two dimensions—having significant implications, especially when considering surfaces or interfaces. In our exercise, when constraining the diffusion of lysozyme to two dimensions, as in a thin layer of water, this changes the calculation for the time it would take for the protein to diffuse an rms distance of 1 μm. Our new computation reveals it would take about 3.85 seconds in such a scenario.

Two-dimensional diffusion is particularly relevant in the study of cell membranes where proteins and lipids diffuse predominantly within the two-dimensional plane of the membrane. Understanding this can lead to insights into various biological signaling pathways and mechanisms of cellular communication.