Bacterial chemotaxis lecture 2 Manipulation & Modeling Genetic manipulation of the system to test the robustness model Explaining Ultrasensitivity and the wide range of sensitivity.

Falk and Hazelbauer (2001) TIBS 26, 257

Questions to address Do we know all the components? Do we know all the biochemical parameters needed for modeling? Can we explain the precision of adaptation vs variation in timing? Can we account for cell to cell variation in chemotaxis? Where does the impressive amplification come from? How do these properties depend on the system parameters? How are flagellar transitions coordinated? How does the motor work? How did this complex and beautiful system evolve

Barkai&Leibler 1997 Naure 387, 913 Simplified adaptation model with the key assumption that CheB can only demethylate the active form of the receptor/kinase complex. In this model, when fewer receptors are active due to an acute increase in ligand binding, CheB has less substrate available so demethylation is slowed while methylation (CheR) is constant resulting in a net increase in receptor methylation over the next few minutes. Since methylation stabilizes the activation state of the receptor (even when ligand is bound), the net activation ultimately returns to the original value. This model argues that availability of “active” receptor substrate for CheB is more important for perfect adaptation than is the phosphorylation state (activity) of CheB.

A st = K b V R max /(V B max -V R max ) Assumptions in Baraki&Leibler adaptation model: Notice that the ability of active receptors to cause phosphorylation of CheB and inactivate CheB is not considered in this model ( V B max is treated as a constant). Thus, this regulation is not required for perfect adaptation. If considered, this regulation would be predicted to increase the steady state fraction of active receptors but the system would still exhibit perfect adaptation. But now, V B max (t) = k phos A(t) - k dephos At steady state, V B max (st) = k phos A st - k dephos Since the system perfectly adapts, V B max (st) is a constant

The model predicts that increasing the amount of CheR (methylase) from 100 to 300 molecules/cell increases the fraction of active receptors at steady state (Activity) resulting in a larger fraction of time spent tumbling and also resulting in a shorter time required to recover following addition of ligand. Barkai&Leibler 1997 Naure 387, Consistent with simplified model A st = K b V R MAX /(V B MAX -V R MAX ) = K b V R MAX /V B MAX for V B MAX >>V R MAX It can also be shown that  1/V R MAX (see Alon Chapter exercises) Thus, substituting for V R MAX,  K b /V B MAX A st

Barkai-Leibler Model is a form of Integral Control Yi, Huang, Simon&Doyle 2000 PNAS 97, 4649 In the classic integral control model (on right), applied to bacterial chemotaxis, u is the fraction of receptors that are not bound to ligand (the external perturbation) and x is the fraction of receptors that are not methylated due to demethylation of active receptors by CheB (the response). y 1 is the resulting activity state of the receptor, which is linearly related to the fraction of receptors unbound minus the fraction of receptors demethylated y 1 = k(u-x) (e.g. methylated and unbound receptors have the highest probability of being active). k is a positive constant relating the total activity of the receptors to the fraction unoccupied and methylated. y 0 is the steady state level of y 1. At steady state, y 1 = y 0 K b V R max /(V B max -V R max ) (from Barkai&Leibler) a function only of CheR and CheB binding and turnover numbers - independent of ligand concentration. y is defined as the difference between the activity at time t (y 1 ) and the activity at steady state (y 0 ). Thus, at steady state, y = 0. Decreased ligand binding acutely increases u and elevates y 1 to a value above y 0, giving a transient positive value for y.

At steady state, (y = 0) the rate of methylation and demethylation are equal. If one assumes for simplicity (as did Barkai&Leibler) that CheR is saturated and unaffected by ligand binding and that the CheB demethylase only acts on active receptors (whether or not ligand is bound) then the net rate of demethylation at any instantaneous time will be directly proportional to y (the transient excess in active receptors over the steady state value). When y = 0 methylation and demethylation cancel out. The fraction of demethylated receptors (x) at any time t is then determined by the number of receptors in the demethylated state at time zero, x 0 (e.g. prior to the unbinding perturbation) plus the number of receptors that get demethylated during the interval in which the system was perturbed. This latter term is the integral from the time at which the perturbation (e.g. ligand unbinding) occurred t=0 to time t of ydt. So x(t) = x 0 + ydt Notice that y can be + or - depending on whether ligand decreases or increases. Thus dx/dt = y = k(u-x) - y 0 At steady state, dx/dt=y=0 and y 1 =y 0 Notice that since k and y 0 are constants, an increase in u (rapid release of ligand) is ultimately offset by a slow decrease in x so that at steady state k(u-x) = y 0. Barkai-Leibler Model is a form of Integral Control Yi, Huang, Simon&Doyle 2000 PNAS 97, t

Assumptions/simplifications in integral model: 1.CheB only acts on active receptors (essential for perfect adaptation with robustness). 2.The activity of the unmethylated receptor is negligible compared to methylated. 3.The binding of methylase CheR to receptors is not affected by ligand binding. 4.The Vmax values of the methylase and demethylase are independent of receptor occupancy or methylation state. Variations from these assumptions compromises perfect adaptation.

The basics of chemotaxis Receptor:CheW:CheA phosphorylates CheY Phosphorylated CheY interacts with motor to promote CW rotation and tumble CheY dephosphorylated by CheZ Attractant binding reduces CheA activity -> less CheY-P -> rarer tumbling Repellent binding increases CheA activity -> more CheY-P -> more tumbling Adaptation via control of methylation: Ligand binding and receptor methylation jointly control CheA activity At given ligand occupancy, more methylation -> more CheA activity At given methylation level, more attractant (less repellent) binding -> less CheA activity CheA phosphorylates and activates CheB, the receptor methylase Attractant -> Less CheA activity -> Less CheB-P -> more methylation ->more CheA activity Repellent -> More CheA activity -> More CheB-P -> Less methylation ->Less CheA activity Effectively the system measures the difference between the extent of two processes: Fast ligand binding Slow receptor methylation/demethylation When [attractant] changes fast, the two signals show a large difference & cells respond

1. A model is only useful if it makes predictions that can be tested by experimentation. The most useful experiments for testing a model involve making quantifiable changes in the concentrations of individual components and monitoring the consequent time-dependent changes in system behavior. 2.Models for biological systems can never be proven. For every simple model it is always possible to imagine a complex model that works equally well. Thus one must always chose the simplest model (fewest parameters) that adequately explains the behavior of the system. Experiments should always be designed to disprove the model, not to support the model.

Experimental data to test robustness of adaptation Alon, Surette, Barkai and Leibler, 1999 Nature 397, 168 Adaptation time for wild type E. coli 1 mM aspartate added at time 0 (open circles) Mock addition (squares). Notice perfect adaptation

Cells lacking CheR do not tumble (triangle). Addition of various levels of CheR back to E. coli lacking the CheR gene results in a hyperbolic increase in the steady state tumbling ratio (consistent with Barkai&Leibler model -equation below). The adaptation time (in response to 1 mM aspartate) decreases with CheR expression (also consistent with model) but the adaptation precision is nearly perfect at all CheR as long as CheR is not zero (upper graph). At low receptor activation, the steady state tumbling rate is proportional to steady state receptor activation (A st ) which is predicted to increase with increasing CheR (V R max in the equation below): A st = K b V R max /(V B max -V R max ) assuming V B max >V R max and other simplifications tumbling  Experimental data to test robustness of adaptation Alon, Surette, Barkai and Leibler, 1999 Nature 397, 168

Fold Expression Effect of varying expression of Che proteins on behavior Experimental data to test robustness of adaptation Alon, Surette, Barkai and Leibler, 1999 Nature 397, 168 Conclusion: In agreement with the Barzai&Leibler model, the precision of adaptation is robust even when Che protein levels vary by an order of magnitude. However, adaptation time and tumbling frequency change considerably.

Ultrasensitivity to a wide range of chemoattractants Bacteria can respond to small differences (<1% front to back) in chemoattractant concentration over a very large (1000 fold) range of basal concentrations. Given what we know about receptor saturation, how can this work? [Attractant] Receptor Occupation A B C D E F Inconsistent with simple hyperbolic or sigmoidal saturation of receptor binding A B C DE F

Controllable Amplification Sourjik, V., et al. (2002). Proc Natl Acad Sci U S A 99: Cells initially at 0, 0.1,.5, 5 mM MeAspAdd indicated [MeAsp] Measure fold decrease in response as a function of the change in [MeAsp] (lower) After adaptation to the new [MeAsp], restore [Asp] to initial value and monitor the increase in tumbling frequency (upper curves) S=27 S=36

Alon et al., 1998 EMBO J. 17, 4238 Increasing the level of phosphorylated-CheY results in increased tumbling frequency

Ultrasensitivity of the flagellar motor to changes in phospho-CheY Cluzel, Surette and Leibler (2000) Science 287, 1652

Fluorescence Correlation Spectroscopy allows a determination of the concentration of a fluorescent molecule (protein), as well as the diffusion rate. See Elson, 2001 Traffic 2, 789 The average fluorescence (over time) detected when a laser beam is focused on a small subcompartment of the cell is given by = Q = Q  where n is the number of molecules in the subcompartment at a particular time,  is the average number of particles (over time) and Q is a constant characteristic of the quantum yield of the fluorophore and the sensitivity of the detection system. Molecules will enter and leave the volume resulting in fluctuations in the fluorescence intensity (see graph A). The probability that there will be n molecules in the defined volume at any particular time can be predicted by the Poisson distribution: P(n) =  n exp(-  )/n! and for this distribution, the variance can be shown to be =  The fluorescence fluctuation autocorrelation function G(t) is defined as the fluctuation of the fluorescence from the average fluorescence at time t multiplied by the fluctuation at time t+  and then averaged over many points on the fluorescence intensity output (Graph A): Fluorescence Intensity G(  ) = Lim N->infinity i=1    F(it)- ][F(it+  )- ] Thus, G(  ) = Q 2 Lim N->infinity i=1    n(it)-  ][n(it+  )-  ] Notice that when  = 0, G(0) = Q 2 = Q 2  If one divides by the square of the average fluorescence intensity, the Q 2 term drops out: this normalized value G(0) is often used to define the autocorrelation function. G(0)/ 2 = G(0) = 1/  Thus, the inverse of G(0) provides the number of molecules in the volume under investigation. It can also be shown that G(t) is related to the diffusion constant D of the fluorescent molecule and the radius of the laser beam by the equation in Graph B. G(t) = 1/{  [1+(4Dt/  2 )]}

Ultrasensitivity of the flagellar motor to changes in phospho-CheY Cluzel, Surette and Leibler (2000) Science 287, 1652 G(t) = 1/{  [1+(4Dt/  2 )]} where  is the number of molecules in the detection volume, D is the diffusion constant and  is the radius of the detection volume. Fluorescence correlation spectroscopy allows a determination of the number of fluorescent molecules in a given volume visualized by a laser beam from the fluctuation in fluorescence intensity in the detection volume with time

Clockwise bias (direction for tumbling) increases in a sigmoid fashion in response to increased CheY-P. The rate of switching from CW to CCW is highest at the CheY-P concentration that gives 50% maximal response. A fit to the data gives a Hill coefficient of ~10 Ultrasensitivity of the flagellar motor to changes in phospho-CheY Cluzel, Surette and Leibler (2000) Science 287, 1652

Alon et al., 1998 EMBO J. 17, 4238 A MWC model for phospho-CheY binding to a cluster of ~30 coupled receptors Assumptions: 1.The receptor cluster exists in two states (favoring CCW or CW rotation of the motor). 2.In the absence of phospho-CheY, the receptor cluster is in a conformation driving CCW rotation (swimming). 3.Binding one phospho-CheY proteins makes the CCW conformation less favorable by a factor of . 4.Saturation of receptors with phospho-CheY strongly stabelizes the CW rotation (tumbling).

Is there cooperativity in receptor inactivation by ligands?

Bray infectivity model to explain ultrasensitivity Bray et al., 1998 Nature 393, 85 To explain the extreme sensitivity of E. coli to chemoattractants, Bray et al., propose that binding of ligand to a single receptor decreases the activation state of a group of adjacent receptors. The change in activation of the entire receptor complex is given by:  P = n A n 1  where n A is the number of receptors bound, n 1 is the number of adjacent receptors affected and  is the change in receptor activation state (p 0 - p 1 ) upon binding to ligand. The fractional saturation of receptors is n A /N = C A /(C A + K d ) The minimal concentration of A that is sufficient to cause a significant change in the receptor activation (  P min ) is then given by: C min A = K d (n min A )/(N-n min A ) = K d (  P min /n 1  )/(N-  P min /n 1  If n1 is larger than 1, then the concentration of ligand required to elicit a response is much lower. However the ability to higher concentrations is impaired.

Raindrop model of receptor saturation To expalin how this system can still respond at high concentrations of ligand, it is assumed that some receptors are not connected to the cluster and respond independently.

Wolnin & Stock (2004) Current Biology 14, R486 Chemoreceptors work in clusters in a cooperative manner to regulate CheY activity

MWC model for chemoreceptor cooperativity Sourjik and Berg, 2004 Nature 428, 437

CheW Receptor trimer of dimers CheA b) Cluster of trimer dimers held together by CheA and CheW Rao, Frenklach and Arkin 2004 JMB 343, Trimers of receptor dimers are the primary unit. 2.Receptor methylation stabilizes the active form. 3.Trimer-dimer aggregate into higher complexes via CheW and CheA. 4.Cooperativity comes from neighboring interactions in the higher complex. 5.The concentration of CheA and CheW determines cluster size and hence degree of cooperativity.

Active StatesInactive States Shaded receptors indicate ligand occupation. In addition to the equilibria shown, each Inactive trimer-dimer is in equilibrium with its analogous Active trimer-dimer of equivalent ligand occupation. Ligand binding shifts the equilibrium to the inactive state while methylation shifts to the active state Rao, Frenklach and Arkin 2004 JMB 343, 291 Assumptions: 1.Receptor methylation increases the change in free energy from A to I (e.g.  G >  G ) 2.Ligand binding makes the free energy change from A to I more negative for any given methylation state. (e.g.  G m 000 >  G m 111 ) 3.For the homo Trimer-Dimer, all forms with a single ligand bound are treated identically in regard to ligand affinity K(A 000 -A 001 ) = K (A 000 -A 010 ) and treated identically for  G from state A to I (for simplicity, noted as  G m 1 ). Analogous assumption for all 2 ligand states (designated  G m 2 ). Methylation state Ligand ocupation state of the three dimers

Ligand concentration should be  M rather than mM Effect of varying the A to I free energy change for 0, 1, 2 or 3 ligand occupation on the ligand- dependent activation state of the homo-Trimer-Dimer (for  G 0,  G 1 and  G 2 variations,  G 3 is fixed at -2.7 kcal/mol, insuring that the triply ligated form is inactive). Notice that increasing  G 1 and  G 2 lowers the apparent affinity and increases the cooperativity.

Fit of Dimer Trimer model to in vitro data for inactivation of purified Asp receptor by methyl- aspartate. (data from Bornhorst and Falke 2000, 2001; model from Rao, Frenklach and Arkin 2004 JMB 343, 291 Mimicking methylation by amino acid substitution of Asp receptors (1, 2, 3, or 4 substitutions) shifts the equilibrium in the absence of ligand toward the active state (  G 0 goes from negative to slight positive. The apparent K d for Asp shifts to a higher value and the Hill coefficient shifts to a slightly higher value (from 1.6 to 2.4). Figure b; A detailed fit of data from Bornhorst and Falke to the Dimer-Trimer model gives values for  G 0,  G 1,  G 2, and  G 3 consistent with expectations. All four  G values become less negative as methylation is increased. The model fits the data well. 97  M 16  M

In vitro data for the Ser receptor requires a model with at least 6 receptors in a cluster (Dimer of Trimer-Dimers). Data from Li and Weiss, While  G 0 changes only marginally with methylation (always between -2 and -1 Kcal/mol), the  G for intermediate levels of ligand occupation are very large negative numbers, indicating that in the absence of methylation, ligand binding to only 1 or 2 of the dimers in the cluster of 6 is enough to shift the entire cluster to the Inactive state, explaining ultrasensitivity at low levels of methylation. At high levels of methylation,  G is less negative for the intermediate states of ligand occupation and response to receptor occupation becomes dramatically less sensitive but much more sigmoidal. This is because at a methylation state of 4,  G 0 through  G 5 are small negative numbers but  G 6 is still a large negative number. Thus, inactivation requires that all 6 sites be occupied (Hill coefficient near 6). 0.2  M 1 mM

Me-Asp (mM)Tar expression (fold normal) Tar +, Tsr + Tap + Tar +, Tap + Tar + Kinase Activity Me-Asp Ser Evidence that distinct chemotactic receptors interact to affect the cooperativity of CheA activation Sourjik and Berg, 2004 Nature 428, 437 In cells expressing only Tar, Me-Asp inhibition of CheA activity is sensitive and cooperative (Hill Coefficient of 3). Wild type levels of Tap expression (a minor receptor) reduces the sensitivity to Me-Asp. Wild type levels of both Tap and Tsr dramatically reduce the sensitivity to Me-Asp and also reduce the cooperativity (Hill coefficient of 1). Increasing the level of Tar expression with constant wild type levels of Tsr allows Me-Asp to completely inhibit CheA kinase activity. Rao, Frenklach and Arkin (2004) successfully fit these data with a mixed Trimer- Dimer model. Since Tsr can bind Me-Asp at very high concentrations, the model can also explain how cells respond to Me-Asp over nearly 6 orders of magnitude.

+Ser For cells expressing both Tar and Tsr, addition of a subsaturating amount of Ser (70  M) increases the sensitivity to Me-Asp. Sourjik and Berg, 2004 Nature 428, 437

Summary Ultrasensitivity is derived by cooperative behavior in phospho-CheY binding to its receptor at the motor, allowing a small change in P-CheY to produce a shift in direction of the motor. Ultrasensitivity is also derived from cooperative behavior in attractant binding to cell surface receptors. In addition, distinct chemoattractant receptors can interact to give synergistic responses between distinct ligands.

Degree of clustering depends on the ratio of Receptor, CheA and CheW Rao, Frenklach and Arkin 2004 JMB 343, 291 CheW CheA Receptor