Diffusion in biological systems and Life at low Reynolds number

Transcription

1 Diffusion in biological systems and Life at low Reynolds number Aleksandra Radenovic EPFL Ecole Polytechnique Federale de Lausanne Bioengineering Institute Laboratory of Nanoscale Biology Lausanne September 26 th 2013.

2 Main contents Diffusion in biological systems Diffusion in the cell Diffusive dynamics Biological applications of diffusion Life at low Reynolds number Friction in fluids Low Reynolds number world Biological applications

3 Brownian motion Brown (1828) Pollen grains (1μm) suspended in water do a peculiar dance The motion of the pollen has never stopped Totally lifeless particles do exactly the same thing Einstein (1905) Quantitative explanation from thermal motion of water molecules

4 Random Walk Models for Life in Motion Dynamics in cells is mostly driven by random jiggling of small molecules in solution: diffusion Because diffusion of molecules is powered by thermal energy, movement is random. Collisions in chemical reactions Ligand binding/unbinding Distribution of ions or newly made proteins Homogeneous distribution of abundant molecules Equilibrium of pressure and equipartition of energy Opening of ion channel & diffusion of ions When channel is open, ions diffusion inward Passive transport (no energy input)

5 Diffusive & directed motions of RNA polymerase Passive: Free RNA polymerase molecule diffusing in a bacterial cell Active: One dimensional motion of RNA polymerase along DNA Energy source: NTP

6 Patterns of E. coli swimming at different scales At low magnification, the swimming movement of a single bacterium appears to be a random walk At higher magnification, it is clear that each step of this random walk is made up of very straight, regular movements

7 Biological distances measured in diffusion times How long does it take to move across the bacterial cell? Diffusion time as a function of the length

8 Diffusion is no effective over large cellular distances Nerve cell In reality, it takes much longer than 12 days, because viscosity of cytoplasm is much higher for large particles and the crowding effect slows down diffusion. Time scale of the active transport by virtue of molecular motors is shortened by many orders of magnitude than passive diffusion

9 Experimental techniques: measuring diffusive dynamics Fluorescence recovery after photobleaching Fluorescently labeled molecule Photobleached region by laser pulse Fluorescently labeled Molecules diffuse into the region

10 Snapshots from photobleaching and photoactivation experiments

11 Snapshots from photobleaching and photoactivation experiments FRAP and photoactivation measurements must be interpreted with caution; in particular, one cannot assign an effective viscosity to the cytoplasm which would be applicable to all proteins inside it. Mobility depends sensitively on the protein under consideration, on its concentration, and on any genetic modification it may have undergone, such as His tagging. Thus, protein mobility must be seen not only as a property of the geometrical structure of cytoplasm and the background macromolecular concentrations alone but also as a characteristic of the diffusing species

12 FRAP experiment showing recovery of a GFP labeled protein confined to the membrane of the endoplasmic reticulum The boxed region is photobleached at time instant, t = 0. In subsequent frames, fluorescent molecules from elsewhere in the cell diffuse into the bleached region.

13 FRAP FRAP denotes an optical technique capable of quantifying the diffusion of fluorescentlylabeled biological molecules. Diffusion can be in 1 D in long processes (e.g. flagellum), 2 D on membrane proteins in a lipid bilayer or 3 D inside cytoplasm. (B) High power laser pulse is sent to photobleach all of the fluorescent probes in certain region. (C) Recovery of the fluorescent from the neighboring area is recorded as a function of time. (D) The half life of recovery is a function of diffusion constant. Recovered signal may be

14 Single particle tracking Record trajectory {x(t),y(t),z(t)} of a particle Calculate Diffusion constant

15 Fluorescence correlation spectroscopy 1) Measure the fluorescence intensity in a small region of the cell as a function of time (2) By analyzing the temporal fluctuations of the intensity through the use of timedependent correlation functions, the diffusion constant and other characteristics of the molecular motion can be uncovered such as concentration

16 Random Walk Redux

17 Diffusive dynamics :Diffusion & Random Walk What is a meaning of <> Equivalent to release simultaneously M particles at origin and then let them do independently random walk (or diffuse to other places)

18 Random Walk Redux Diffusion can be modeled as one, two or three dimensional random walk: Step size is fixed Number of steps is not fixed and time dependent Assume that the molecule takes one step at every Δt. From one dimensional random walk: D is an experimentally measurable parameter

19 D is an experimentally measurable parameter We assumed molecular scale parameters L and Δt, which are not experimentally measurable. To verify the idea that diffusion and friction are manifestations of thermal motion, Einstein noticed that there is a third relation between Land Δt.

20 Diffusive dynamics :Fick's law Assumption Large number particles Uniform in y, z direction Nonuniform in x direction Jump distance L per time Δt Same jump probability to left and right N(x,t): the number of particles in the box between x L/2 and x+l/2 at time t. (1/2)N(x,t) particles will pass through x L/2 to the left in Δt. Simultaneously, (1/2)N(x L,t) particles pass through x L/2 to the right Concentration of molecules

21 Diffusive dynamics :Fick's law Question: why is there a flux since each particle has the same jump probability to left and right? Reason for net flow: there are more particles in one slot than in the neighbouring one due to nonuniform distribution. Continuity equation

22 Diffusive dynamics :Fick's law Average rate of crossing a surface per unit are is flux J measures the number of particles moving from left to right (+x direction) If there are more particles on the left, c is decreasing by x, and its negative derivative is positive. There will be net flux of particles towards right. Particles will move from densely populated regions to sparsely populated regions to equilibrate their chemical potential (Entropic Forces)

23 Diffusion Equation If there is order in the initial state, diffusion will erase that memory. Fick s Law is useful if concentration gradient is time independent. However if the concentration gradient is changing over time due to diffusion of particles, Fick s Law does not tell much. N(x,t)

24 Fundamental Pulse Equation What happens if we introduce N number of particles to x = 0 position in pure water. Net flux of particles on the left towards the left side Net flux of particles on the right towards the right side Concentration of particles decreases by time due to diffusion of particles away from 0 position Diffusion finally erases the bump. We expect the variance of the pulse to increase with time. Can simple Gaussian be a solution?

25 Fundamental Pulse Equation

26 Biased Diffusion A force F exerted on a particle results in a drift velocity: The movement of particles under the same net v drift would generate a net flux: We can write the total flux due to random diffusion and net movement under the drift velocity:

27 Biased Diffusion In equilibrium, the net flux vanishes, J(x) = 0 Out of equilibrium,

28 The Role of Diffusion in Biological Reactions The cell membrane is uniformly decorated a by receptor proteins. A signaling process depends on the arrival of ligand molecules and attachment to receptors. The spherical cell of radius R is introduced into a solution with concentration c 0 at the far distance. We assume that the cell membrane is a perfect absorber, that concentration of ligand at the surface is 0. In a steady state

29 The Role of Diffusion in Biological Reactions

30 The Role of Diffusion in Biological Reactions What if the receptors have a finite rate of kon n adsorbing ligands? The number of adsorbed ligands per unit time, where M is the number of receptors on the membrane:

31 The Role of Diffusion in Biological Reactions

32 Universal Rate for Diffusion Limited Chemical Reactions dn 8 We obtain k diff dt kt 3

33 Friction in Fluids We learned that diffusion is a dissipative process. It tends to erase ordered arrangements of molecules. Friction is dissipative, too. It tends to erase ordered motion. In biology, dynamics of small molecules can be envisioned as a marble dropped into a jar of honey. We will learn that water acts like a highly viscous fluid for small molecules. Viscous friction dominates mechanics in the nanoworld. We learned that v drift F small particles fall under gravity because the nominator (F) increases by the mass (m) of particle which is proportional to R 3, whereas the drag coefficient (ς) increases by R. Therefore vdrift is proportional to R 2. How about small particles?

34 Pulse Equation under Bias You release 1 billion molecules at position x = 0 in the middle of a narrow tube. The molecules, diffusion constant is 100 μm2s 1. An electric field pulls the molecules to the right with drift velocity of 1 μms 1. After 80s, what percentage of molecules will be on the left?

35 Pulse Equation under Bias

36 Pulse Equation under Bias

37 The Role of Diffusion in Biological Reactions

38 Fluctuation Molecular thermal motion (random) ==> Diffusion equation Question: Why is diffusion equation deterministic? Answer: Average on ensemble (collection of large number repeated systems) available to describe a system of large number particles. Finite particle system: its behavior deviated from the prediction of diffusion equation. This deviation is called statistical fluctuation. Few particle system: Statistical fluctuations will be significant, and the system's evolution really will appear random, not deterministic. Nanoworld of single molecules: need to take fluctuations seriously

39 Stokes Drag at a Single Molecule Level ATP synthase is a rotary motor which turns 120 counterclockwise per ATP produced. To demonstrate the rotation of the enzyme, fluorescently tagged actin filament was attached to the F1 subunit Myosin V motor carries a micron sized bead in optical trap experiments, demonstrating its ability to carry large vesicles and organelles. Viscous drag produced in these experiments is: Drag force is small compared to force production of a single motor (2 6 pn)

40 Friction in fluids Suspension

41 Small Particles Remain in Suspension in Water Most proteins have slightly higher densities than water. Gravity pulls the object downward. Buoyant force reduces the effect of gravity.

42 Centrifugation While the molecules stay homogenously distributed under Earth s gravity, they can be forced to sediment under centrifugal acceleration with many orders of magnitude of g. The particle wants to move in a straight line (no acceleration). The centripetal acceleration points toward the axis of rotation. This force can only come from frictional drag force of the surrounding fluid. Therefore, particle remains in solution, while slowly drifting towards the bottom of the tube. Centrifugation artificially increases the gravity.

43 Friction in fluids Centrifuge Centripetal

44 Friction in fluids Sedimentation time scale

45 Separation of Molecules by Ultracentrifugation Under centrifugation, molecules both undergo diffusional motion and drifted motion. Each species of particles will have different vdrift and D, depending its density and size. Width of the band of molecules will increase by diffusive law x 2Dt The center position of the band will be given by x v t Center drift Since particle moves linearly with time, by dissipate by the square root of time, the method allows separation of molecules.

46 Friction in fluids Sedimentation time scale

47 Friction in fluids Hard to mix a viscous liquid

48 Friction in fluids Hard to mix a viscous liquid

49 Low Reynolds Number World Newtonian fluid

50 Low Reynolds Number World Viscous critical force

51 Low Reynolds Number World Newtonian fluid

52 Low Reynolds Number World Reynolds number Osborne Reynolds FRS (23 August February 1912)

53 Low Reynolds Number World Newtonian fluid

54 Sedimentation Centrifugation can be used to separate particles with different sizes from each other.

55 Viscosity of Fluids, revisited The response of a fluid to applied force is flow. If you rapidly pull a spoon from a jar of honey, the whole jar will come along. If you pull slowly, there will be little force resisting it. Viscosity is a mechanical property of a fluid, resisting motion. (Right) When you move one plate with force F, it shears the hypothetical layers of water. The force will create a linear velocity gradient from the top to the bottom plate. Velocity is equal to zero at the bottom plate surface.

56 Shearing of Water Shear Modulus of Solids Viscosity of fluids Solids shear under force and come to an equilibrium. Fluids flow under force. The unit of viscosity is Pa.s Velocity is inversely proportional to viscosity. Shear Modulus of Solids Viscosity of Fluids

57 Laminar and Turbulent Flow When a fluid encounters an obstacle, it flows the way around it. If the viscosity of the fluid is high, and velocity is low, it will display a laminar flow. In contrast, fluid with low viscosity forms turbulent flow at high velocity.

58 Reynolds Number Assume that motion of a small fluid element, a cube with a side length of l, encounters an obstacle with a diameter of R. To sidestep the sphere, fluid must undergo radial acceleration Newton s Law of motion says that To estimate the frictional force, we need to generalize Newtonian Fluid relation where the velocity of fluid is not uniform

59 Reynolds Number The net frictional force on a fluid is the force exerted above it, minus the force exerted below it. The Reynolds number is the ratio of inertial and friction terms Note that R ~ l, length scale is similar to the size of the molecule.

60 Reynolds Number In other words, kinetic energy of a particle travelling with a velocity v: Energy dissipated by viscous friction: Ratio of these terms again gives the Reynolds number.

61 The Life at Low Reynolds Number When Re is small, friction term dominates. When Re is large, inertial term dominates. In the low Re world, momentum (or kinetic energy) is rapidly dissipated by friction. In other words, when a ship stops its engines, it still moves a long distance (due to its inertia) before it stops. When a fish swims in water, friction has some effect on its motion. Fish can move a distance similar to its size without swimming before it stops completely. When a bacteria stops its engines, it stops moving in 0.1 Å. Water acts like an intensely viscous environment

62 Dissipative Time Scales and the Reynolds Number We can estimate the time to dissipate kinetic energy of the molecule in a fluid. Time it takes a molecule to move comparable to its size: The ratio of these two terms again yields the Reynolds number (R ~ l) Dissipation of kinetic energy by viscous drag: At times greater than τ viscous, the inertial term can be dropped in above equation. Dependence of the velocity of the object on its initial velocity also drops out if viscous time scale is shorter than time it takes to travel a distance of x(0). Without bias, swimming objects lose their momentum within the dissipative timescale

63 How Do Small Organisms Swim? Suppose you flap a paddle, then bring it back to its original position by the same path. No net progress! Reciprocal motion won t work for swimming. It must be periodic so that it can be repeated. Swimming cycle can be divided into two strokes that are time reversals of one another. Scallop model (three plates and two hinges) has been proposed to explain swimming motility of Spiroplasma. NYvQTWYc

64 Bacterial Flagella Generates Net Force in Direction of Motion Tangential velocity per small section of flagellum experiences a drag force that resists its motion. Viscous drag coefficient for motion parallel to the axis is SMALLER than the one in perpendicular axis of motion. Therefore, force vector is not parallel to the velocity vector, and is closer to the perpendicular direction. x and y components of force cancel out (symmetry), but the one in z does not.

65 Bacterial Flagella Generates Net Force in Direction of Motion

66 Bacterial Flagella Generates Net Force in Direction of Motion velocity of bacteria is 30 μm/sec. Why does bacteria swim? If the food is plenty, diffusion is sufficient to feed the bacteria. It is only relevant, when diffusion takes much longer, or food is rare. Bacteria moves in a straight line, pauses, and takes of in a new random direction. While swimming, cell continuously sample the environment for rich nutrients. Biased random walk

67 Further reading Further reading Reichl, A modern course in statistical physics Phillips, Physical biology of the cell, Ch13 Nelson, Biological physics, Ch4