Practice: Molecules in motion

You watch a 1-µm bead jiggle in water under a microscope. In 5 separate 1-second intervals, it moves these distances along the x axis: 0.7, −1.2, 0.8, −0.6 and −0.8 µm. What is the mean (average) displacement, in µm? Keep the signs: a minus means a move to the left.

5 walkers each take 36 random steps of +1 or −1, flipping a coin for each step. They end at −12, 12, −2, 0 and 4 steps from where they started. What is the mean square displacement: the average of the squared distances, in steps²?

Many walkers each take 400 random steps of +1 or −1. The mean square of their end positions is 400 steps². What is the root mean square (RMS) distance from the start, in steps?

A random walker should end up typically 10 times as far from its start as before (same step length). How many times as many steps does it need?

In syn3A's 3D model, the cell is split into 8-nm boxes and a molecule moves by hopping from a box to a neighbor. After 100 hops, how far is a molecule typically from where it started, in nm?

A protein's motion is drawn as 1-nm hops in random directions. About how many hops does it take to get typically 50 nm from its start?

GFP in water diffuses with D = 87 µm²/s. What is its mean square displacement (MSD) after 1 ms, in 3D? Give it in µm².

GFP in water diffuses with D = 87 µm²/s. An older table lists values in cm²/s. What is it in cm²/s?

In an experiment, GFP molecules in water are followed in 3D. Their mean square displacement after 1 ms is 0.53 µm². What is their diffusion coefficient D, in µm²/s?

GFP diffuses in water with D = 87 µm²/s. For GFP, the cytoplasm of a mammalian cell behaves like a fluid 3.2 times as viscous as water (Swaminathan and colleagues, 1997). What D would you expect for GFP there, in µm²/s?

Use Stokes–Einstein, $D = kT / (6\pi\eta r)$, for a GFP-size protein in water at 20 °C, where the viscosity is η = 1.002 mPa·s. Take r = 2.8 nm and k = $1.381 \times 10^{-23}$ J/K. What is D, in µm²/s?

A protein diffuses for 9 times as long as before. How many times as far from its start does it typically get?

A ribosome in E. coli diffuses with D = 0.04 µm²/s. About how long does it take to wander across syn3A, about 0.4 µm? Use $t = x^2/(6D)$ and give the answer in ms.

A GFP-size protein in nerve cytoplasm diffuses with D = 27 µm²/s. About how long would it take to wander 1 cm along an axon, a long thin tube where it can only move along the line? Use $t = x^2/(2D)$ and give the answer in days.

A ribosome in E. coli diffuses with D = 0.04 µm²/s. How far from its start does it typically get in 1 second, in 3D? Give the answer in nm.

A human cell is about 20 µm across. Compare two ways to move a big complex the size of a ribosome (D = 0.04 µm²/s) across it: diffusion, with $t = x^2/(6D)$, or riding a kinesin motor at 0.8 µm/s. Whichever is faster, how many times longer does the slower way take?

Over a distance x, diffusion takes about $x^2/(6D)$ and a kinesin motor at v = 0.8 µm/s takes $x/v$. For GFP in mammalian cytoplasm (D = 27 µm²/s), at what distance do the two take the same time? Give it in µm.

syn3A has a volume of about 0.0335 µm³. A protein is at 5 µM in it. About how many molecules of it does the cell hold?

A large syn3A cell in the same tomograms had a radius of 247 nm and held 820 ribosomes. What percent of the cell's volume do the ribosomes fill, treating each as a sphere of radius 10 nm?

syn3A (0.0335 µm³) holds about 503 ribosomes. If they were spread evenly, about how far apart would neighboring ribosomes be (center to center)? Use $(V/N)^{1/3}$ and give it in nm.

One protein (D = 7.7 µm²/s) searches for one partner; they react when they touch, at a distance R = 5 nm. The mean search time in a volume V is $t = V/(4\pi DR)$. In what volume would the search take 1 s on average? Give it in µm³.

A less stable mRNA (made up) has a half-life of 2 minutes. What is the rate constant k, per minute?

A syn3A cell holds 500 copies of a protein and stops making new ones. They break down with k = 0.0277 per hour (a 25-hour half-life). So $dA/dt = -kA$. What is dA/dt at the start, in proteins per hour?

A box holds 1,000 protein complexes, which fall apart with k = 10 per second (made-up numbers). So $dA/dt = -kA$. Take one Euler step of Δt = 0.01 seconds. How many complexes does Euler's method predict after the step?

A syn3A cell holds 8 copies of an mRNA and stops making new ones. Each copy breaks down with k = 0.1733 per minute (a 4-minute half-life). Use Euler's method with steps of Δt = 0.5 minutes. Starting from 8, what value do you get after 4 steps (at t = 2 minutes)?

A box holds 1,000 protein complexes, which fall apart with k = 0.001 per second (made-up numbers). So $A(t) = A_0 e^{-kt}$. On average, how many complexes are left after 1,000 seconds?

A test tube holds 100 molecules of A, which break down with k = 0.5 per second. Euler's method with steps of Δt = 1 second runs to t = 2 seconds. By how much does it fall short of the exact answer $A_0 e^{-kt}$ there?

For the reaction A → B, mass action says the rate is $k[\text{A}]$, in M/s (molar per second), with each concentration in M. The units of k are $\text{M}^{n}\,\text{s}^{-1}$. What is n?

In A + B → C (rate $k[\text{A}][\text{B}]$), both [A] and [B] go up 10 times. How many times as fast is the reaction?

A + B → C has k = $10^{6}$ M⁻¹ s⁻¹. With [A] = 2 µM and [B] = 3 µM, how fast is C made? Give the rate in µM/s.

Two proteins A and B react with k = $10^{6}$ M⁻¹ s⁻¹. In syn3A (volume 0.0335 µm³) there is one A and one B. On average, how long until they meet and react? Give it in s.

A + B → C has k = $10^{5}$ M⁻¹ s⁻¹. Think of one A molecule among 1,000 B molecules in syn3A (volume $3.35 \times 10^{-17}$ L). On average, how long until the A reacts? Give it in ms.

If every meeting reacts, the fastest possible rate constant is the Smoluchowski limit, $k = 4\pi(D_A + D_B)(R_A + R_B) \times N_A \times 1{,}000$, with D in m²/s and R in m. Work it out for two GFP-size proteins in water at 37 °C, each with D = 117 µm²/s and radius 2.8 nm. Give k in M⁻¹ s⁻¹.

The pair in the Particles vs Equations simulation binds with kon = $10^{7}$ M⁻¹ s⁻¹ and comes apart with koff = $10$ s⁻¹. What is its dissociation constant Kd, in µM?

In living E. coli, a lac repressor (LacI) stays on its DNA operator for about 5 minutes on average before it comes off (Hammar and colleagues, 2014). What koff does that mean, in s⁻¹?

A cell holds 150 nM of a partner protein in all, but 50 nM of it is already tied up in complexes. Kd = 100 nM. What percent of the time is the protein bound? Use fraction bound = L/(L + Kd), where L is the free partner's concentration.

A protein binds a partner with Kd = 1 µM. How much free partner is needed for the protein to be bound 99% of the time? Give it in µM.

A pair binds with Kd = 10 nM. In syn3A (volume $3.35 \times 10^{-17}$ L), how many free copies of the partner make a concentration equal to Kd?

A human cell is about 20 µm across. Compare two ways to move a protein as fast as GFP in E. coli (D = 7.7 µm²/s) across it: diffusion, with $t = x^2/(6D)$, or riding a kinesin motor at 0.8 µm/s. Whichever is faster, how many times longer does the slower way take?

A small syn3A cell in a cryo-electron tomogram (Gilbert and colleagues, 2021) had a radius of 201 nm and held 503 ribosomes. What percent of the cell's volume do the ribosomes fill, treating each as a sphere of radius 10 nm?

Glucose in water diffuses with D = 600 µm²/s. About how long does it take to wander across a human cell, about 20 µm? Use $t = x^2/(6D)$ and give the answer in ms.

In an experiment, ribosomes in E. coli are tracked in a flat (2D) microscope image. Their mean square displacement after 5 seconds is 0.81 µm². What is their diffusion coefficient D, in µm²/s?

A + B → C has k = $10^{7}$ M⁻¹ s⁻¹. With [A] = 100 nM and [B] = 100 nM, how fast is C made? Give the rate in nM/s.

A protein's motion is drawn as 1-nm hops in random directions. About how many hops does it take to wander typically 400 nm, the width of syn3A?

A pair binds with kon = $10^{5}$ M⁻¹ s⁻¹ and koff = $0.5$ s⁻¹ (made-up numbers). In syn3A (volume $3.35 \times 10^{-17}$ L), how many free copies of the partner make a concentration equal to Kd?

If every meeting reacts, the fastest possible rate constant is the Smoluchowski limit, $k = 4\pi(D_A + D_B)(R_A + R_B) \times N_A \times 1{,}000$, with D in m²/s and R in m. Work it out for two proteins with syn3A's model value, D = 1.0 µm²/s each, and radius 2.8 nm. Give k in M⁻¹ s⁻¹.