Lecture 35: Diffusion & Fluorescence Recovery After Photobleaching

Diffusion lecture plan Diffusion of single particles as a random walk Parallels first lac operon lecture Adds introduction to the diffusion equation Diffusion as a consequence of chemical potential differences Alternative derivation of Fick’s second law Description of the diffusion coefficient Fluorescence Recovery After Photobleaching Example applications Diffusion to detection (Berg & Purcell)

What have we learned so far? Potentials have associated forces Chemical potential is associated with the driving force of diffusion

What have we learned so far? Diffusing particles are impeded by drag from collisions with solvent molecules The drag coefficient reflects the viscosity (h) of the medium and the shape of the particle

What is the net effect of both forces? As a diffusing particle is accelerated by Fm, it gains velocity. Fdrag grows in magnitude as the particle’s velocity grows. Eventually, Fdrag = -Fm, so Fnet = Fdrag + Fm = 0. Since Fnet = ma, the particle stops accelerating: it has reached terminal velocity vT. Sign indicates that the force will point in the opposite direction of the velocity: that is, it well act against its motion.

How fast is a diffusing particle going when it reaches terminal velocity? You may not believe right now that You’ll have to wait a few slides to be convinced that this D is equivalent to our old diffusion coefficient. Notice that it depends on shape and temperature.

Ultimately I want to show you that the concentration c(x,t) obeys Fick’s second law. Time derivative of c: the rate at which particles flow into a region minus the rate at which they flow out Sample question we could answer if we knew the flux: How many sugar molecules per second flow through a cell’s surface? The first step will be calculating the flux (flow rate per unit area), J

What is the flux (flow rate per unit area) of particles due to diffusion? We will calculate the number of particles that pass rightwards through the plane A in a small time Dt. We will divide that number by A Dt. Our answer will have units of molecules per unit time per unit area (flux). Important to say: even though this is a 3D volume, we’ll assume that the concentration gradient only varies in one direction (x).

What is the flux (flow rate per unit area) of particles due to diffusion? During a short time Dt, diffusing particles move a distance of: All particles up to a distance Dx away pass through the plane A in time Dt.

What is the flux (flow rate per unit area) of particles due to diffusion? The # of particles crossing the plane rightwards is Therefore, the flux J is: If Delta n is negative, this means the particles tend to cross leftwards instead. If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

What have we learned by calculating the flux? Particles are flowing in the direction in which the concentration is decreasing In general there is a net flow of particles, so the concentration profile is changing with time: can we figure out how? If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

How does the concentration profile change with time? How many particles go into this box in time Dt? How many leave? Net change in particle #? If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

How does the concentration profile change with time? What is the net change in concentration? If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

How does the concentration profile change with time? In the limit of very small times Dt, we can approximate these ratios by derivatives: If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

But this probably does not surprise you! The concentration profile evolves in time in the same way that a single particle’s probability distribution evolves! If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense. But this probably does not surprise you!

Principle of superposition If f(x,t) and g(x,t) are solutions to a diffusion equation, then h(x,t) = c1 f(x,t) + c2 g(x,t) is, too! If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

If f(x,t) and g(x,t) are solutions to a diffusion equation, then h(x,t) = c1 f(x,t) + c2 g(x,t) is, too! The solution h(x,t) is considered “more general”.

Principle of superposition If f(x,t) and g(x,t) are solutions to a diffusion equation, then h(x,t) = c1 f(x,t) + c2 g(x,t) is, too! Each diffusing particle obeys Fick’s second law, so a group of many particles must also obey Fick’s second law. If the slope of the concentration is positive (i.e. it’s increasing), then the flux is going the other way (toward lower concentration) – makes good sense.

Diffusion lecture plan Diffusion of single particles as a random walk Parallels first lac operon lecture Adds introduction to the diffusion equation Diffusion as a consequence of chemical potential differences Alternative derivation of Fick’s second law Description of the diffusion coefficient Fluorescence Recovery After Photobleaching Example applications Diffusion to detection (Berg & Purcell)

Reminder: Fluorescence Fluorescent proteins and dyes absorb photons of one wavelength, and quickly re-emit photons of a (usually longer) wavelength Typical uses of GFP: create a fusion protein

Accidental photobleaching Fluorescence fades after long-term exposure to the excitation beam (e.g. during timecourses). This reflects rare, but irreversible and thus cumulative, damage to fluorophores. The fluorophore is the part that is absorbing the light and re-emitting it. Major damage is from chemical rearrangement that affects not just the fluorescent protein but others in the cell. Here, these HeLa cells are dying.

Why is fluorescence recovered (sometimes)? Photobleaching can also be done on purpose with an intense, narrow beam Endoplasmic reticulum membrane protein vs. nuclear laminar protein. Important to note that the beam is shined brightly but very briefly. Why is fluorescence recovered (sometimes)?

Fluorescence “recovers” when non-bleached molecules diffuse into the bleached region Or more slowly as new fluorescent protein is made. You can see that the “dark” molecules also diffuse out and after equilibrium is reaches, the whole cell is a little darker. (Not noticeably so if only a small region was bleached.)

Fluorescence Recovery After Photobleaching (FRAP) is used to measure diffusion coefficients

The diffusion coefficient’s value can tell us: Whether a molecule/protein is confined Bound to something rigid In a compartment Tethered Whether a protein is in a complex Larger molecules have more drag, smaller D Might reflect dimerization, binding, aggregation What medium a protein is diffusing through Viscosity is larger (i.e. drag is greater) for diffusion through membrane bilayers than through cytoplasm

Model: perfect photobleaching boundary Before Photobleaching After Photobleaching

New math trick: convolution This animation is not just to shock you into complacency so that you let me introduce math again – I swear.

Introduction to Pyrotechnics To design a firework show, you only need to decide when and where a firework will go off. The firework will do the rest: it knows how to spread out in space and get brighter, then dimmer, with time.

Finding c(x,t) from c(x,t=0) Our pyrotechnician-like approach: Decide how many particles to “deploy” between x and x+dx at time t=0, in order to get the desired c(x,t=0) Add up all of their probability distributions Done!

Finding c(x,t) from c(x,t=0) How many particles need to be placed between position k and position k + dk at time t=0? c(x=k,t=0) dk = c(k,0) dk How do we make a particle start at an arbitrary position x=k (instead of the origin)? Shift it! Starts at x=0 Starts at x=k

Finding c(x,t) from c(x,t=0) Now we just have to sum up over all positions: Probability distribution for a particle starting at x=k Number of particles starting at x=k

Finding c(x,t) from c(x,t=0)

Finding c(x,t) from c(x,t=0)

Finding c(x,t) from c(x,t=0)

Finding c(x,t) from c(x,t=0) Notice that this integral is basically the area under the curve for a portion of a Gaussian.

Finding c(x,t) from c(x,t=0) How fast this

Finding c(x,t) from c(x,t=0) FYI, there is no closed-form solution, so to make the plots, I expressed c(x,t) in terms of the error function

Building a FRAP gap using superposition We have effectively been studying one boundary of a region that has been photobleached. Let’s complete the FRAP gap using superposition. How fast this

Building a FRAP gap using superposition How fast this

Building a FRAP gap using superposition Now if we measured the fluorescence profile of a cell after photobleaching, I know it should be of this form, and I can try to fit a value of D to my data.

What we learned today We can find solutions for groups of particles from the single-particle solution using the principle of superposition For example, we can understand how fluorescence should recover after photobleaching This allows us to determine the diffusion coefficient, to learn about protein mobility