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**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

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**Cells know where to go Wild-type leukocyte response to fin wound**

Matthias et al (2006)

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**Close-up view of chemotaxis**

Dictybase Website Cell moves about one cell diameter per minute Decision-making maintains flexibility

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**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.

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**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

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**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

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**New uncaging technique**

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

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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

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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

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Sensing model

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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

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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)

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**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

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**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)

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**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)

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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

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**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)

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**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

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**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)

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**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

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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

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**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

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**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

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**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

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**Experiment vs theory Levine, Kessler and Rappel, 2006**

Simplest prediction: no change in PH-CRAC at back membrane Janetopoulos et al 2004 Background subtraction

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**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

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**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

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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.

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**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

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**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

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**Traction Microscopy Data from del Alamo (Lasheras/Firtel) 2008**

Pole forces at front and rear of cell Motility cycle - Protrusion, contraction

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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

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**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)

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**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)

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**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

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**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

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**Results Displacement max ~ .2microns Stress field Peak ~ 40 Pascal**

Agrees reasonably well with traction microscopy data

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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

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**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

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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

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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!

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