statistical analysis and stochastic modelling of cell migration and bumblebee...
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Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Statistical analysis and stochastic modellingof cell migration and bumblebee foraging
Rainer Klages
School of Mathematical Sciences, Queen Mary University of London
Workshop on Aggregation, Inference and Rare EventsUniversity of Warwick, 18 May 2012
Cell migration and bumblebee foraging 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 stochastic modeling and summary
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Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Part 1:
Cell Migration
Cell migration and bumblebee foraging 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)
Cell migration and bumblebee foraging Rainer Klages 4
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Our cell types and how they migrate
MDCK-F (Madin-Darby canine kidney) cells
two types: wildtype (NHE+) and NHE-deficient (NHE−)
movie: NHE+: t=210min, dt=3min
Cell migration and bumblebee foraging Rainer Klages 5
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Measuring cell migration
Cell migration and bumblebee foraging Rainer Klages 6
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?
but: cell migration is active motion, not passively driven!
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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
Cell migration and bumblebee foraging Rainer Klages 8
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
Cell migration and bumblebee foraging Rainer Klages 9
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
Cell migration and bumblebee foraging Rainer Klages 10
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 v2
th = kT/m and Riemann-Liouville fractional(generalized ordinary) derivative of order 1 − αfor α = 1 Langevin’s theory of Brownian motion recovered
• analytical solutions for msd(t) and P(x , t) can be obtainedin terms of special functions (Barkai, Silbey, 2000; Schneider,Wyss, 1989)
• 4 fit parameters vth, α, κ (plus another one for short-timedynamics)
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Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Possible physical interpretation
Physical meaning of the fractional derivative?
the 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 generatesthe same msd(t) and vac(t) as the fractional Klein-Kramersequation
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
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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,...)
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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
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Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Part 2:
Bumblebee Foraging
(PhD project of Friedrich Lenz)
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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.
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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
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Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Variation of the environmental conditions
safe and dangerousflowers
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
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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?
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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
Cell migration and bumblebee foraging Rainer Klages 20
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!
Cell migration and bumblebee foraging Rainer Klages 21
Outline Cell migration Results Summary Bumblebee foraging Results Summary Conclusion
Velocity autocorrelation function ‖ to the wall
V ACy (τ) =
〈(vy (t)−µ)(vy (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 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
correlations change from positive (spider-free) tonegative (spiders)
⇒ all changes are in the velocity correlations , not in pdf’s!
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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
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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
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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)
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