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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 Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Outline
two parts:
1 cell migration
2 bumblebee foraging
in both cases:
motivation and experiment
experimental results and statistical analysis
theoretical modeling and summary
Statistical physics of biological motion Rainer Klages 2
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Part 1:
Cell Migration
Statistical physics of biological motion Rainer Klages 3
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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)
Statistical physics of biological motion Rainer Klages 4
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Why cell migration?
motion of the primordium in developing zebrafish:
Gilmour (2008)
positive aspects:
morphogenesis
immune defense
negative aspects:
tumor metastases
inflammation reactions
Statistical physics of biological motion Rainer Klages 5
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 6
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 7
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Measuring cell migration
Statistical physics of biological motion Rainer Klages 8
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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?
Statistical physics of biological motion Rainer Klages 9
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 10
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 11
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 12
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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)
Statistical physics of biological motion Rainer Klages 13
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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)
Statistical physics of biological motion Rainer Klages 14
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 15
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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,...)
Statistical physics of biological motion Rainer Klages 16
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 17
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Part 2:
Bumblebee Foraging
Statistical physics of biological motion Rainer Klages 18
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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.
Statistical physics of biological motion Rainer Klages 19
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 20
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 21
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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?
Statistical physics of biological motion Rainer Klages 22
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 23
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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!
Statistical physics of biological motion Rainer Klages 24
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 25
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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!
Statistical physics of biological motion Rainer Klages 26
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 27
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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
Statistical physics of biological motion Rainer Klages 28
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
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)
Statistical physics of biological motion Rainer Klages 29