MAE 545: Lecture 2 (9/22) E. coli chemotaxis

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1 MAE 545: Lecture 2 (9/22) E. coli chemotaxis

Recap from Lecture 1 Fokker-Planck equation 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` x In general the probability distribution of jump lengths s can depend on the particle position x.

2 Recap from Lecture 1 Fokker-Planck equation 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` x In general the probability distribution of jump lengths s can depend on the particle position x. Assume that jumps occur in regular small time intervals (s x) apple applev(x)p(x, @x 2 D(x)p(x, t) drift velocity (external fluid flow, external potential) v(x) = X (s x) =hs(x)i st t s diffusion coefficient (e.g. position dependent temperature) D(x) = X s 2 s 2 2 t (s x) = (x) 2 t s 2

Lévy flights Probability of jump lengths in D dimensions

better strategy than random walk for finding prey that is

3 Lévy flights Probability of jump lengths in D dimensions Normalization condition Moments of distribution p( ~s ) = Z C ~s, 0, ~s > s0 ~s < s0 dd ~s p( ~s ) = 1 h~si = 0 ~s = 3.5, D = 2 >D 2 = AD s20, 1, >D+2 <D+2 Lévy flights are better strategy than random walk for finding prey that is scarce D. W. Sims et al. Nature 451, (2008) 3 2D random walk

4 Number of distinct sites visited by random walk Total number of sites inside explored region after N steps 1D 2D N tot / p N N tot / N In 1D and 2D every site gets visited after a long time 2r / p N 3D N tot / N p N In 3D not every site is visited even after a very long time! Shizuo Kakutani: A drunk man will find his way home, but a drunk bird may get lost forever. Number of distinct visited sites after N steps 4 1D 2D 3D N vis p 8N/ N vis N/ ln(8n) N vis 0.66N

5 Fick s laws Adolf Fick 1855 N noninteracting Brownian particles Local concentration c(x, t) =Np(x, t) Fick s laws below follow from Fokker-Plank equations First Fick s law Flux of particles J = Diffusion of particles Second Fick @x apple @x 5

6 First Fick s laws Estimate flow of particles due to concentration gradient N(x) N(x + x) At next time step half of particles jump to the left and half to the right x x + x J = 1 2 [N(x + x) N(x)] t J = D 1 x Net flux x2 x 2 = x 2 2 t 1 x 6 apple N(x + x) [c(x + x) c(x)] x N(x) x In the presence of flow we need to add transport of molecules J =

7 Second Fick s laws Estimate change in concentration due to gradient in flow N(x, t) J(x, t) J(x + x, t) x x + x 1 t x [N(x, t + t) N(x, t)] = 1 t x 1t [c(x, t + t) c(x, t)] = 1 x [J(x + x, t) J(x, t)] t [J(x + x, t) @x 7

8 Fick s laws Adolf Fick 1855 N noninteracting Brownian particles Local concentration c(x, t) =Np(x, t) Fick s laws below follow from Fokker-Plank equations First Fick s law Flux of particles J = Diffusion of particles Second Fick @x apple @x Generalization to higher dimensions ~J = ~vc = ~ r (~vc)+ ~ r 2 (Dc) 8

9 E. coli chemotaxis L. Turner, W.S. Ryu, H.C. Berg, J. Bacteriol. 182, (2000) 9

10 Escherichia coli 0.8µm 2.5µm flagella E. coli is a part of gut flora that helps us digest food. Concentration of E. coli Total concentration of bacteria 10 9 cm cm 3 10 In normal conditions E. coli divide and produce 2 daughter cells every ~20min. In one day one E. coli could produce ~7x10 10 new cells!

11 Flagella filaments and rotary motors length L. 10µm Flagellum filament left handed helix helix diameter d 0.4µm filament diameter 20nm pitch p 2.3µm Rotary motor 45nm 11

12 Swimming of E. coli F thrust F drag swimming speed v s 20µm/s body spinning frequency f b 10Hz spinning frequency of flagellar bundle f r 100Hz Thrust force generated by spinning flagellar bundle F thrust = F drag 6 Rv s F thrust 0.4pN = N Torque generated by spinning flagellar bundle N = N drag 8 R 3! b N 2pN µm = Nm size of E. coli water viscosity R 1µm 10 3 kg m 1 s 1 12

13 How quickly E. coli stops if motors shut off? F drag swimming speed v s 20µm/s F thrust size of E. coli R 1µm water viscosity 10 3 kg m 1 s 1 mass of E. coli m 4 R3 3 4pg Newton s law ma = 6 Rv x = x 0 h 1 e t/ i E. coli stops almost instantly! m 6 R 2 R2 9 x 0 = v s 0.1Å 0.2µs signature of low Reynolds numbers Re = Rv s

14 Translational and rotational diffusion x hx 2 i =2D T t 2 =2D R t Einstein - Stokes relation Einstein - Stokes relation D T k BT 6 R 0.2µm2 /s D R k BT 8 R rad2 /s After ~10s orientation changes by 90 degrees due to Brownian motion! size of E. coli R 1µm water viscosity temperature Boltzmann constant 10 3 kg m 1 s 1 T = 300K k B = J/K 14

15 E. coli chemotaxis Rotary motor 45nm typical duration: 15 Run swimming speed: v s 20µm/s t r 1s all motors turning counter clockwise Increase (Decrease) run durations, when swimming towards good (harmful) environment. Tumble random change in orientation h i = 68 typical duration: t t 0.1s one or more motors turning clockwise

16 E. coli chemotaxis ˆn ẑ Homogeneous environment run duration: t r 1s tumble duration: t t 0.1s swimming speed: v s 20µm/s drift velocity v d =0 D e = D e effective diffusion `2 6 h ti v2 st 2 r 6(t r + t t ) 60µm2 /s 16 Gradient in food concentration run duration increases (decreases) when swimming towards (away) from food t r (ˆn) =t r + (ˆn ẑ)(@c/@z) v d = h h drift velocity zi ti v s (@c/@z) 3( t r + t t )

$Average fraction of bound receptors pb is related to concentration c of molecules. p B = c c 0 = k o c + c 0 k on Singling network inside E.$

17 Sensing of environment E. coli surface is covered with receptors, which can bind specific molecules. Average fraction of bound receptors pb is related to concentration c of molecules. p B = c c 0 = k o c + c 0 k on Singling network inside E. coli analyzes state of receptors and gives direction to rotary motor. k on k o 17

18 Diffusion limited flux of molecules to E. coli Fick = Dr2 c = D 1 boundary conditions c(r!1)=c 1 c(r) =0 absorbing sphere steady state c(r) =c 1 apple1 R r J(r) = flux of rate of absorbing molecules = Dc 1R r 2 I(r) =J(r) 4 r 2 = 4 DRc 1 = I 0 = k on c 1 diffusion constant for D 10 3 µm 2 /s k on 10 4 µm 3 /s small molecules I = N absorbing disks of radius s I 0 1+ R/Ns example R 1µm 18 s 1nm flux drops by factor 2 for N = R/s 3000 fractional area covered by these receptors (N s 2 )/(4 R 2 ) 10 3 E. coli can use many types of receptors specific for different molecules, without significantly affecting the diffusive flux

19 Accuracy of concentration measurement How many molecules do we expect inside a volume occupied by E. coli? Probability p(n) that cell measures N molecules follows Poisson distribution p mean N p(n) = N N E N N! Error in measurement Err N N (R3 c) 1/2 N R 3 c standard N N = deviation for c =1µM = m 3 ) Err 4% E.coli can be more precise by counting molecules for longer time t. However, they need to wait some time t0 in order for the original molecules to diffuse away to prevent double counting of the same molecules! t 0 R 2 /D 10 3 s N R 3 ct/t 0 DRct Err (DRct) 1/2 for t=1s, precision improves to Err~0.1% When E. coli is swimming, it wants to swim faster than the diffusion of small molecules v s t & (Dt) 1/2 ) t & D/v 2 s 1s 19

20 How E. coli actually measures concentration? Probability for motor to rotate in CCW direction (runs) as a function of time in response to short pulse in external molecular concentration Input concentration 1s 3s E. coli integrates measured concentration observed during the last second and compare this with measured concentration during the previous 3 seconds. If difference is positive then increase the probability of runs, otherwise increase the probability of tumbles. J. E. Segall, S. M. Block, and H. C. Berg, PNAS 83, (1986) 20

Adaptation Probability for motor to rotate in CCW direction (runs) as a function of time in response to a sudden increase in external molecular concentration 1-0F Input concentration (n.2 0.

21 Adaptation Probability for motor to rotate in CCW direction (runs) as a function of time in response to a sudden increase in external molecular concentration 1-0F Input concentration (n CD Time (sec) E. coli adapts to the new level of concentration in about 4 seconds. This enables E. coli to be very sensitive to changes in concentration over a very broad range of concentrations! J. E. Segall, S. M. Block, and H. C. Berg, PNAS 83, (1986) 21

How efficient is motor of E. coli? Energy source for rotary motor are charged protons ph =7.0 H + Each proton gains energy due to Transmembrane electric potential difference 120mV Change in ph U =( 2.

22 How efficient is motor of E. coli? Energy source for rotary motor are charged protons ph =7.0 H + Each proton gains energy due to Transmembrane electric potential difference 120mV Change in ph U =( 2.3k B T/e) ph 50mV Total protonmotive force p = + U 170mV Need 1200 protons per one revolution Input power P in = e p f = 170meV 10Hz pn nm/s ph 7.8 H + Power loss due to stokes drag P rot = N (2 f) 4600pN nm (20 Hz) pn nm/s P trans = F v 0.4pN 20000nm/s pn nm/s 22 Motor efficiency P trans + P rot P in 90%

23 Further reading 23

Bacterial Chemotaxis

Bacterial Chemotaxis Bacteria can be attracted/repelled by chemicals Mechanism? Chemoreceptors in bacteria. attractant Adler, 1969 Science READ! This is sensing, not metabolism Based on genetic approach!!!