Statistical physics of biological motion: Crawling cells ...klages/talks/biomot_ucalg.pdf · Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion Statistical

Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion

Statistical physics of biological motion:Crawling cells and foraging bumblebees

R. Klages

School of Mathematical Sciences, Queen Mary University of London

Complexity Science GroupDepartment of Physics and Astronomy

University of Calgary, 13 April 2012

Statistical physics of biological motion Rainer Klages 1


Outline

two parts:

1 cell migration

2 bumblebee foraging

in both cases:

motivation and experiment

experimental results and statistical analysis

theoretical modeling and summary



Part 1:

Cell Migration



Brownian motion of migrating cells?

Brownian motion

Perrin (1913)three colloidal particles,positions joined by straightlines

Dieterich et al. (2008)single biological cell crawling ona substrate

Brownian motion?conflicting results:yes: Dunn, Brown (1987)no: Hartmann et al. (1994)



Why cell migration?

motion of the primordium in developing zebrafish:

Gilmour (2008)

positive aspects:

morphogenesis

immune defense

negative aspects:

tumor metastases

inflammation reactions



How do cells migrate?

membrane protrusions andretractions ∼ force generation:

lamellipodia (front)uropod (end)actin-myosin network

formation of a polarized statefront/end

cell-substrate adhesion



Our cell types and some typical scales

renal epithelial MDCK-F (Madin-Darby canine kidney) cells;two types: wildtype (NHE+) and NHE-deficient (NHE−)

observed up to 1000 minutes: here no limit t → ∞!

cell diameter 20-50µm; mean velocity ∼ 1µm/min;lamellipodial dynamics ∼ seconds

movies: NHE+: t=210min, dt=3min NHE-: t=171min, dt=1min



Measuring cell migration



Theoretical modeling of Brownian motion

‘Newton’s law of stochastic physics’ :

v̇ = −κv+√

ζ ξ(t) Langevin equation (1908)

for a tracer particle of velocity v immersed ina fluid

force decomposed into viscous damping andrandom kicks of surrounding particles

Application to cell migration?



Mean square displacement

• msd(t) :=< [x(t) − x(0)]2 >∼ tβ with β → 2 (t → 0) andβ → 1 (t → ∞) for Brownian motion; β(t) = d ln msd(t)/d ln t

anomalous diffusion if β 6= 1 (t → ∞); here: superdiffusion



Velocity autocorrelation function

• vac(t) :=< v(t) · v(0) >∼ exp(−κt) for Brownian motion• fits with same parameter values as msd(t)

crossover from stretched exponential to power law



Position distribution function

• P(x , t) → Gaussian(t → ∞) and kurtosis

κ(t) := <x4(t)><x2(t)>2 → 3 (t → ∞)

for Brownian motion (greenlines, in 1d)

• other solid lines: fits fromour model; parameter valuesas before

note: model needs to beamended to explainshort-time distributions

100

10-1

10-2

10-3

10-4

10 0-10

p(x,

t)

x [µm] 100 0-100

x [µm] 200 0-200

x [µm]

OUFKK

100

10-1

10-2

10-3

10-4

10 0-10

p(x,

t)

x [µm] 100 0-100

x [µm] 200 0-200

x [µm]

OUFKK

2 3 4 5 6 7 8 9

500 400 300 200 100 0

kurt

osis

Κ

time [min]

a

b

c

NHE+t = 1 min t = 120 min t = 480 min

NHE-t = 1 min t = 120 min t = 480 min

data NHE+

data NHE-

FKK model NHE+

FKK model NHE-

crossover from peaked to broad non-Gaussian distributions



The model

Fractional Klein-Kramers equation (Barkai, Silbey, 2000):

∂P∂t

= − ∂

∂x[vP] +

∂1−α

∂t1−ακ

[

∂

∂vv + v2

th∂2

∂v2

]

P

with probability distribution P = P(x , v , t), damping term κ,thermal velocity vth and Riemann-Liouville fractional derivativeof order 1 − α defined by

∂γP∂tγ

:=

{

∂mP∂tm , γ = m∂m

∂tm

[

1Γ(m−γ)

∫ t0 dt ′ P(t ′)

(t−t ′)γ+1−m

]

, m − 1 < γ < m

with m ∈ N; for α = 1 ordinary Klein-Kramers equationrecovered

4 fit parameters vth, α, κ (plus another one for ‘biological noise’on short time scales)



Solutions for this model

analytical solutions (Barkai, Silbey, 2000):

mean square displacement:msd(t) = 2v2

tht2Eα,3(−κtα) → 2Dαt2−α

Γ(3−α) (t → ∞)

with Dα = v2th/κ and generalized Mittag-Leffler function

Eα,β(z) =∑∞

k=0zk

Γ(αk+β) , α , β > 0 , z ∈ C;

note that E1,1(z) = exp(z): Eα,β(z) is a generalizedexponential function

velocity autocorrelation function:vac(t) = v2

thEα,1(−κtα) → 1κΓ(1−α)tα (t → ∞)

for κ → ∞ fractional Klein-Kramers reduces to a fractionaldiffusion equation yielding P(x , t) in terms of a Foxfunction (Schneider, Wyss, 1989)



Possible physical interpretation

Physical meaning of the fractional derivative?

fractional Klein-Kramers equation is approximately related tothe generalized Langevin equation

v̇ +∫ t

0 dt ′ κ(t − t ′)v(t ′) =√

ζ ξ(t)

e.g., Mori, Kubo (1965/66)

with time-dependent friction coefficient κ(t) ∼ t−α

cell anomalies might originate from glassy behavior of thecytoskeleton gel, where power law exponents are conjecturedto be universal (Fabry et al., 2003; Kroy et al., 2008)

nb: anomalous dynamics observed for many different cell types



Possible biological interpretation

Biological meaning of the anomalous cell migration?

experimental data and theoretical modeling suggest slowerdiffusion for small times while long-time motion is faster

compare with intermittent optimal search strategies of foraginganimals (Bénichou et al., 2006)

note: controversy about modeling the migration of foraginganimals (albatros, bumblebees, fruitflies,...)



Summary: Anomalous cells

different cell dynamics on different time scales(cp. with Lévy hypothesis, which suggests scale-freeness)

for long times cells crawl superdiffusively with power lawdecay of velocity correlations and non-Gaussian positionpdfs

stochastic modeling of experimental data by a generalizedKlein-Kramers equation



Part 2:

Bumblebee Foraging



Motivation

bumblebee foraging – two verypractical problems:

1. find food (nectar, pollen) incomplex landscapes

2. try to avoidpredators

What type of motion?

Study bumblebee foraging in a laboratory experiment.



The bumblebee experiment

Ings, Chittka, Current Biology 18, 1520 (2008):bumblebee foraging in a cube of ≃ 75cm side length

artificial yellow flowers: 4x4 grid onone wall

two cameras track the position(50fps) of a single bumblebee(Bombus terrestris)

advantages: systematic variation of the environment;easier than tracking bumblebees on large scales

disadvantage: no ‘free flight’ of bumblebees



Variation of the environmental conditions

safe and dangerousflowers

movie

three experimental stages:

1 spider-free foraging

2 foraging under predation risk

3 memory test 1 day later

#bumblebees=30 , #data per bumblebee for each stage ≈ 7000



Bumblebee experiment: two main questions

1 What type of motion do the bumblebees perform in termsof stochastic dynamics?

-0.1 0

0.1 0.2

0.3 0.4

0.5 0.6 -0.2

-0.1 0

0.1 0.2

0.3 0.4

0.5 0.6

-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

z

x y

z

2 Are there changes of the dynamics under variation of theenvironmental conditions?



Velocity distributions: analysis

0.01

0.1

1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

ρ(v y

)

vy [m/s]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Qua

ntile

s of

the

data

[m/s

]

Quantiles of PDF with parameters estimated from data [m/s]

left: experimental pdf of vy -velocities of a single bumblebee inthe spider-free stage (black crosses) with max. likelihood fits ofmixture of 2 Gaussians; exponential; power law; singleGaussian

right: quantile-quantile plot of a Gaussian mixture against theexperimental data (black) plus surrogate data



Velocity distributions: interpretation

best fit to the data by a mixture of two Gaussians withdifferent variances (quantified by information criteria withresp. weights)

biological explanation: models spatially different flightmodes near the flower vs. far away, cf. intermittentdynamics

big surprise: no difference in pdf’s between differentstages under variation of environmental conditions!



Velocity autocorrelation function ⊥ to the wall

V ACx (τ) = 〈(vx (t)−µ)(vx (t+τ)−µ)〉

σ2 with average over all bees

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

v xac

(τ)

τ [s]

0.001

0.01

0.1

0.1 1 10

P(T

f)

Tf [s]

plot: spider-free stage, predation thread, memory test∃ anti-correlations for τ ≃ 0.5: bees return to flowersonly small quantitative changes under predation thread,cf. shift of minimum in V AC

x (τ) and changes in pdf of flighttimes (inset): more flights with long durations



Velocity autocorrelation function ‖ to the wall

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

v yac

(τ)

τ [s]

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1 1.5 2

plot: spider-free stage, predation thread, memory test∃ profound qualitative change of correlations frompositive for spider-free to negative in case of spidersresampling of data (inset) confirms existence of positivecorrelations

⇒ all changes are in the velocity correlations , not the pdf’s!



Predator avoidance and a simple model

predator avoidance asdifference in position pdfsspider / no spider from data:

-0.06-0.03

0 0.03

0.06

yrel [m] 0

0.04

0.08

zrel [m]

-0.03-0.02-0.01

0 0.01 0.02 0.03

∆ρp(xrel,yrel)

positive spike: hovering;negative region: avoidance

modeled by Langevin equationdvydt (t) = −ηvy (t) − ∂U

∂y (y(t)) + ξ(t)

η: friction coefficient,ξ: Gaussian white noise

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

v yac

(τ)

τ [s]

simulated velocity correlations withrepulsive interaction potential Ubumblebee - spider off / on



Summary: Clever bumblebees

mixture of two Gaussian velocity distributions reflectsspatial adjustment of bumblebee dynamics to flower carpet

all changes to predation thread are contained in thevelocity autocorrelation functions , which exhibit highlynon-trivial temporal behaviour

(nb: Lévy hypothesis suggests that all relevant foraginginformation is contained in pdf’s)

change of correlation decay in the presence of spiders dueto experimentally extracted repulsive force asreproduced by generalized Langevin dynamics



Collaborators and literature

work performed with:

1. cells: P.Dieterich, R.K., R.Preuss, A.Schwab,Anomalous Dynamics of Cell Migration, PNAS 105, 459 (2008)

2. bees: F.Lenz, T.Ings, A.V.Chechkin, L.Chittka, R.K.,Spatio-temporal dynamics of bumblebees foraging underpredation risk, Phys. Rev. Lett. 108, 098103 (2012)


Documents

Statistical physics of biological motion: Crawling cells ...klages/talks/biomot_ucalg.pdf · Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion Statistical