1. Signal versus Noise in Eukaryotic Chemotaxis
Herbert Levine
UCSD-CTBP (an NSF-supported PFC)
With: Wouter-Jan Rappel, W. Loomis, D. Fuller (Biology)
W. Chen, B. Hu, M. Buenemann, D. Shao (CTBP)
Outline:
Experiments on chemotactic response in Dictyostelium
Signal versus noise in gradient sensing
Nonlinear amplification via signal transduction
Cell motility mechanics

2. Cells know where to go Wild-type leukocyte response to fin wound
Matthias et al
(2006)

3. Close-up view of chemotaxis
Dictybase Website
Cell moves about one cell diameter per minute
Decision-making maintains flexibility

4. Cell migration in a gradient
cAMP gradient
flow rate 640 m/s
1h real time =
8 sec movie
tD = l2/D
D= (kT)/(6πha)
a: size of theparticle
h: viscocity of the fluid
T: absolute temp
a ~ (MW)1/3
Re=rul/h At low Re, viscous drag dominates.
u: fluid flow speed,
L: length scale (channl diameter)
r: density
h: viscocity
Laminar flow is smooth predictable flow which always occurs at Re <<1, where any induced vortices die away.

5. Detection threshold at ~3x10-3 nM/micron
Track multiple cell positions and determine chemotactic velocity
vx: velocity in flow direction
vy: velocity perp. to flow direction vy vx
Detection threshold at ~3x10-3 nM/micron
This translates into an occupancy difference of ~50 receptors with an occupancy mean of 300 (in the middle of the device)
Song et al., 2006

6. Subcellular polarity markers
Starting with the work of Parent and Devreotes, response can be tracked by subcellular markers
Uniform CAR1 receptor
Lipid modification via PI3K
F-actin at the front (via RAC)
Myosin at the rear
These allow for the measurement of the kinetics of the gradient sensing step
Receptors
Occupancy
PI3K
PTEN
RAC
Myosin

7. New uncaging technique
From C. Beta, Potsdam
Uncaged cAMP in a flow creates well-defined excitation
Anal Chem. (2007)

8. Questions for theory
Can one predict macroscopic measures of cell motility given the space-time course of an applied signal and (possibly) the cell history
Cell speed, directional persistence, chemotactic index
Not just averages, but also distributions
Not just wild-type, but also mutant strains
We are nowhere near this goal
New pathways still being discovered
Many unresolved questions as to mechanisms

9. Sensing Noise
Sensing is done roughly 50K receptors, each with a binding constant of 30nM.
One can calculate directly the amount of information available to the cell regarding the external gradient angle 
where y is vector of occupancies whose probability distribution is a Gaussian with mean and variance
For small gradients,
Note: Result is in bits of information
Diffusive correlations are negligible

10. Sensing model

11. Sensing Noise
Sensing is done roughly 50K receptors, each with a binding constant of 30nM.
One can calculate directly the amount of information available to the cell regarding the external gradient angle 
where y is vector of occupancies whose probability distribution is a Gaussian with mean and variance
For small gradients,
Note: Result is in bits of information
Diffusive correlations are negligible

12. Compare to experiment
We can directly compare this number to the information about the gradient indicated by the actual cell motion
no gradient
5% gradient
C.I. = ± 0.04
C.I. = 0.556
100m
100m
Movement up gradient
Fuller, Chen et al, PNAS (2010)

13. Result: The cell operates at the sensor noise limit for small
We also analyzed the
instantaneous angle of the
cells in the devices.
In shallow gradients the E + I mutual information
is limited by the external mutual information
(receptor occupancy).
E + I mutual information decrees
at higher concentrations due to
an increase in internal noise.
Result: The cell operates at the sensor noise limit for small
gradients and concentrations; eventually limited by other noise
and/or processing losses

14. Phenomenological approach
We can compare the results of the experiment to a phenomenological model where extra noise is added to the sensing noise
= gradient strength
= noisy estimate of gradient direction
=extrinsic noise (due to bare motility)
From this, we can directly calculate the stationary distribution of angles of cell motion; generalization of circular normal distribution (Hu et al, PRE 2009)

15. Aside: Cooperative receptors
Receptor cooperativity can lower the noise for concentrations close to Kd
Sets bound on cell size for spatial sensing (Hu et al, PRL 2010)

16. Signaling models
Can we go beyond this phenomenological approach to discuss actual gradient sensing process
We will focus on a gradient sensing approach, which tries to explain how external signals can get amplified to the level of decisions

17. LEGI -first conceptual model
Local activation and global inhibition
explains adaptation to global stimulus
versus steady gradient response
Successes
Reasonable match to data (esp. in Latrunculin treated cells)
Shortcomings
Gives linear amplification (x3 in Lat)- no polarity formation!
Inconsistent with data (Postma et al) on post-adaptation structures
Local activation cannot really work in the presence of noise (later)

18. LEGI - simplest version
Since A and I are both proportional to S in steady-state, uniform S results in a transient activation of E but eventual perfect adaptation. With a non-uniform S, I gives average value and A remains local - pattern in the effector E

19. Mathematical aside: Phase Field Approach:
How can we handle complex, deforming domains
Membrane fields can also be defined in a natural way
For stationary shapes: second equation drops out
Boundary conditions are implemented correctly (in the limit of vanishing interface thickness)
Kockelkorn, Rappel +HL PRE (2003)

20. LEGI Local activation and global inhibition
explains adaptation to global stimulus
versus steady gradient response
Successes
Reasonable match to data (esp. in Latrunculin treated cells)
Can be extended to model more biological detail
Shortcomings
Gives linear amplification (x3 in Lat)- no polarity formation!
Inconsistent with data (Postma et al) on spontaneous structures

21. Can we post-amplify?
To amplify the internal gradient, we need to set a threshold for some process
Small gap between front and back at a variable PIP3 level - in simplest model, would need some sort of cell-by-cell learning
Amplification possible via molecular depletion (Nossal et al) - but, this leads to timescale issues
Would fall apart without absolutely perfect adaptation

22. Spontaneous PH-domain localization (Van Haastert et al)
Recent work has pointed out that there is not perfect adaptation to uniform stimulation
Localized PH-domain patterns from in second phase of excitation; very dependent on cell condition
Inconsistent with LEGI model mechanism

23. A decision-making module
Inhibitor acts via removing the activator.
We balance the system such that the amount of (diffusing) inhibitor created by the signal roughly equals the amount of (local) activator - operating guess - trimeric G protein subumits
With gradient: surplus activator in front (follows external signal), surplus inhibitor in back (no response at all!)
Can work this out analytically in a simplified 1d geometry
Membrane-bound activator
Diffusing inhibitor

24. Balanced inactivation model (PNAS 2006)
New model includes a membrane-bound activator, A, a cytosolic inhibitor, B, along with its membrane-bound version Bm
Note balance in
A and B production
Front to back ratio is large if decay rates are small

25. Experiment vs theory Levine, Kessler and Rappel, 2006
Simplest prediction: no change in PH-CRAC at back membrane
Janetopoulos et al 2004
Background subtraction

26. Initial symmetric response - becomes asymmetric after 10 secs
Janetopoulos et al
PNAS (2004)
Xu et al (2005)(NB Latrunculin)
Initial symmetric response - becomes asymmetric after 10 secs
Firtel lab UCSD
System maintains ability to respond to changing gradients

27. Mutual Information after processing
Preliminary results
Model captures high concentration saturation
Possible need for new mechanism at low C
Can be compared to other models (ultrasensitive LEGI)
Background c = 5nm

28. Motility Aspects
The cell needs to take the input sensing data and make a decision, based on cellular signaling networks, which way to go. After this decision, the cell needs to organize itself so as to generate the forces and shape changes needed for the actual motion.
In the remainder of this talk, I will briefly mention our very new efforts to try to understand some of these mechanical aspects of directed cell motility.

29. Phenomenological model of motility (Inbal)
Cell in a microfluidic device with chemoattractant gradient, variable vertical height.
Most of the cell in focal plane: no bleaching issues
Visualize Ras*, an upstream signaling component
Monica Skoge, Loomis lab

30. SIMPLE PHENOMENOLOGICAL MODEL
Model has two components:
A “biochemical” model which is able to produce regions of elevated concentration of a component (patches).
A mechanical model which deforms the cell based on the patch concentration
Tip-splitting; expt. versus simulation
Hecht et al submitted

31. Traction Microscopy Data from del Alamo (Lasheras/Firtel) 2008
Pole forces at front and rear of cell
Motility cycle -
Protrusion, contraction

32. Adhesion sites
Uchida and Yumura showed that that there were specific adhesion sites connecting the actin cytoskeleton to the substratum
These sites come and go as the cell goes through its motility cycle
Our model focuses on the forces generated by these points, modeled as breakable springs
Note: Dicty does not have specialized focal adhesions

33. Dictyostelium cells have (non-specific) adhesion sites
Vertically restricted Dictyostelium cell in gradient, with actin marker limE at the top (green) and at the bottom surface (red)

34. Modeling assumptions (Buenemann et al Biophys J. 2010)
We focus on the part of the cell in contact
The motility cycle consists of contraction (a-c) and then protrusion (c-e)
Contraction is assumed to happen at constant speed, up to a fixed percentage (accomplished by Myosin motors)
During contraction, sites can detach if the force is too high
(Buenemann et al Biophys J. 2010)

35. Adhesive springs Uniform contraction implemented incrementally
Compute forces (rigid or elastic substratum)
After each time-step, adjust cell c.o.m/angle to give zero net force
Allow springs to break at rate
After contraction is done, assume that the cell protrusion carries the cell forward such that its back is at the last adhesive site.
Measure traction forces, cell speeds etc as a function of the parameters

36. Modeling assumptions We focus on the part of the cell in contact
The motility cycle consists of contraction (a-c) and then protrusion (c-e)
Contraction is assumed to happen at constant speed, up to a fixed percentage (accomplished by Myosin motors)
During contraction, sites can detach if the force is too high

37. Results Displacement max ~ .2microns Stress field Peak ~ 40 Pascal
Agrees reasonably well with traction microscopy data

38. Cell velocity
Speed is relatively insensitive to the values of the adhesion parameters, within a significant range
Agrees roughly with some mutant experiments; different than mammalian cells.
But, this model does not have any serious treatment of protrusion
To go further, we need to start thinking about cell deformations and actin-based propulsion

39. Deforming Cells surface bending
We have begun the task of constructing a model of the mechanics of deformation
Our approach is based on a phase-field formulation of the membrane energy coupled to actin-polymerization forces
Surface energy
Bending energy
Area constraint
surface
bending
This formulation can allow us to reproduce results on the
equilibrium shapes of vesicles

40. Preliminary results
To get to moving cells, we need to add in external forces on the protruding front, generated by actin polymerization onto a network attached by adhesions to the substrate; do this is in an ad-hoc fashion
Look at simple cells
Keratocytes
Steady motion
Velocity related to aspect ratio
But, we need to improve our modeling approach

41. Summary
Chemotactic response requires a sophisticated biophysics approach
Models are necessarily spatially-extended
Noise is not negligible
Cell geometry is complex and changeable
We have used a variety of methods to look initially at the signaling and more recently at the mechanics
Eventually, we will understand how it really works!