Near-Perfect Adaptation in Bacterial Chemotaxis Yang Yang and Sima Setayeshgar Department of Physics Indiana University, Bloomington, IN 5/10/2019 Yang Yang, Candidacy Seminar

Intro slide from SS

Intro slide from SS

E. coli and Bacteria Chemotaxis http://www.rowland.harvard.edu/labs/bacteria/index_movies.html Increasing attractants or Decreasing repellents 5/10/2019 Yang Yang, Candidacy Seminar

Chemotaxis Signal Transduction Network in E. coli Pathway Motor Response [CheY-P] Stimulus Flagellar Bundling Motion With approximately 50 interacting proteins , the network converts an external stimulus into an internal stimulus which in turn interacts with the flagella motor to bias the cell’s motion. CheA: Histidine kinase CheB-P: Methylesterase CheW: Couples CheA to MCPs CheY-P: Response regulator CheR: Methyltransferase CheZ: Dephosphorylates CheY-P It is used as a well-characterized model system for the study of properties of (two-component) cellular signaling networks in general. Histidine kinase Methylesterase Couples CheA to MCPs Response regulator Methyltransferase Dephosphorylates CheY-P CheB CheA CheW CheZ CheR CheY Run Tumble 5/10/2019 Yang Yang, Candidacy Seminar

Robust Perfect Adaptation From Sourjik et al., PNAS (2002). Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation) Precision of adaptation insensitive to changes in network parameters (robustness) Adaptation Precison FRET signal [CheY-P] CheR fold expression Fast response Slow adaptation From Alon et al., Nature (1999). 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar This Work: Outline New computational scheme for determining conditions and numerical ranges for parameters allowing robust (near-)perfect adaptation in the E. coli chemotaxis network Comparison of results with previous works Extension to other modified chemotaxis networks, with additional protein components Conclusions and future work 5/10/2019 Yang Yang, Candidacy Seminar

E. coli Chemotaxis Signaling Network Ligand binding Methylation Phosphorylation phosphorylation methylation Ligand binding Chemotaxis in E. coli involves temporal measurement of the change in concentration of an external stimulus. This is achieved through the existence of fast and slow reaction time scales, in the chemotaxis signal transduction network: fast measurement of the current external concentration is compared with the cell’s “memory” of the concentration some time ago to determine whether to extend a run in a given direction or to tumble, thereby randomly selecting a new direction. E=F(free form), R(coupling with CheR), B(coupling with CheBp) E’=F(free form), R(coupling with CheR) 𝜆=o(ligand occupied), v(ligand vacuum) 𝛾=u(unphosphorylated), p(phosphorylated) 5/10/2019 Yang Yang, Candidacy Seminar

Michaelis-Menten Kinetics Enzymatic reaction: Where E is the enzyme, S is the substrate, P is the product. A key assumption in this derivation is the quasi steady state approximation, namely that the concentration of the substrate-bound enzyme changes much more slowly than those of the product and substrate. Therefore, it may be assumed that it is in steady state: Define E, S, P Write final rate equation for P in terms of K_m where Km is the Michaelis Menten Constant (MM constant) 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Reaction Rates 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Approach … START with a fine-tuned model of chemotaxis network that: reproduces key features of experiments is NOT robust AUGMENT the model explicitly with the requirements that: steady state value of CheY-P values of reaction rate constants, are independent of the external stimulus, s, thereby explicitly incorporating perfect adaptation. : state variables : reaction kinetics : reaction rates : external stimulus 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Augmented System The steady state concentration of proteins in the network satisfy: The steady state concentration of = [CheY-P] must be independent of stimulus, s: where parameter allows for “near-perfect” adaptation. Reaction rates are constant and must also be independent of stimulus, s: Discretize s in range {slow, shigh} 5/10/2019 Yang Yang, Candidacy Seminar

Physical Interpretation of Parameter, : Near-Perfect Adaptation Measurement of c = [CheY-P] by flagellar motor constrained by diffusive noise Relative accuracy*, Signaling pathway required to adapt “nearly” perfectly, to within this lower bound (*) Berg & Purcell, Biophys. J. (1977). : diffusion constant (~ 3 µM) : linear dimension of motor C-ring (~ 45 nm) : CheY-P concentration (at steady state ~ 3 µM) : measurement time (run duration ~ 1 second) 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Implementation Use Newton-Raphson (root finding algorithm with back-tracking), to solve for the steady state of augmented system, Use Dsode (stiff ODE solver), to verify time- dependent behavior for different ranges of external stimulus by solving: 5/10/2019 Yang Yang, Candidacy Seminar

Converting from Guess to Solution Starting from initial guess A, the solution to B is generated. A 3%<<5% 1%<<3% 0%<<1% Inverse of T3 MM constant (K3R-1) B Need large legend for blue, green, red points T3 autophosphorylation rate (k3a) 5/10/2019 Yang Yang, Candidacy Seminar

Parameter Surfaces 0%<<1% 1%<<3% Surface 2D projections Inverse of T1 methylation MM constant (K1R-1) Inverse of T1 demethylation MM constant(k1B-1) Need some plots of *slices*, as opposed to projections (which combine all slices in a given direction) How do the slices differ from the pair-wise plots where two parameters only are varied? Inverse of T1 methylation MM constant (K1R-1) 1%<<3% 0%<<1% T1 autophosphorylation rate K1a 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Validation Verify steady state NR solutions dynamically using DSODE for different stimulus ramps: Concentration (µM) Time (s) 5/10/2019 Yang Yang, Candidacy Seminar

Violating and Restoring Perfect Adaptation Inverse of T3 MM constant (K3R-1) CheYp Concentration (µM) T3 autophosphorylation rate (k9) Time (s) Step stimulus from 0 to 1e-3M at t=500s 5/10/2019 Yang Yang, Candidacy Seminar

Conditions for Perfect Adaptation: Kinetic Parameters 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Methylation MM Constant Autophosphorylation Rate Inverse of T0 MM constant (K0R-1) Inverse of T1 MM constant (K1R-1) T0 autophosphorylation rate (k0a) T1 autophosphorylation rate (k1a) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Methylation MM Constant Autophosphorylation Rate Inverse of T2 MM constant (K2R-1) Inverse of T3 MM constant (K3R-1) T2 autophosphorylation rate (k2a) T3 autophosphorylation rate (k3a) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Methylation MM Constant Autophosphorylation Rate Inverse of LT0 MM constant (K0LR-1) Inverse of LT1 MM constant (K1LR-1) LT0 autophosphorylation rate (k0al) LT1 autophosphorylation rate (k1al) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Methylation MM Constant Autophosphorylation Rate Inverse of LT2 MM constant (K2LR-1) Inverse of LT3 MM constant (K3LR-1) LT2 autophosphorylation rate (k2al) LT3 autophosphorylation rate (k3al) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Demethylation MM Constant Autophosphorylation Rate Inverse of T1 MM constant (K1B-1) Inverse of T2 MM constant (K2B-1) T1 autophosphorylation rate (k1a) T2 autophosphorylation rate (k2a) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Demethylation MM Constant Autophosphorylation Rate Inverse of T3 MM constant (K3B-1) Inverse of T4 MIM constant (K4B-1) T3 autophosphorylation rate (k3a) T4 autophosphorylation rate (k4a) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Demethylation MM Constant Autophosphorylation Rate Inverse of LT1 MM constant (K1LB-1) Inverse of LT2 MM constant (K2LB-1) LT1 autophosphorylation rate (k1al) LT2 autophosphorylation rate (k2al) 5/10/2019 Yang Yang, Candidacy Seminar

Inverse of Demethylation MM Constant Autophosphorylation Rate Inverse of LT3 MM constant (K2LB-1) Inverse of LT4 MM constant (K3LB-1) LT3 autophosphorylation rate (k12) LT4 autophosphorylation rate (k13) 5/10/2019 Yang Yang, Candidacy Seminar

Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant T1 demethylation catalytic rate T1 methylation catalytic rate T2 methylation catalytic rate T2 demethylation catalytic rate 5/10/2019 Yang Yang, Candidacy Seminar

Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant T3 demethylation catalytic rate T2 methylation catalytic rate T3 methylation catalytic rate T4 demethylation catalytic rate 5/10/2019 Yang Yang, Candidacy Seminar

Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant LT1 demethylation catalytic rate LT0 methylation catalytic rate LT1 methylation catalytic rate LT2 demethylation catalytic rate 5/10/2019 Yang Yang, Candidacy Seminar

Methylation Catalytic Rate/ Demethylation Catlytic Rate = Constant LT3 demethylation catalytic rate LT2 demethylation catalytic rate LT3 demethylation catalytic rate LT4 demethylation catalytic rate 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Summary The Inverse of Methylation MM constants linearly decrease with Autophosphorylation Rates The Inverse of Demethylation MM constants linearly increase with Autophosphorylation Rates The ratio of Methylation catalytic rates and demethylation catlytic rates for the next methylation level is constant for all methylation states Slides 32 and 33 are redundant These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state receptor model*. * B. Mello et al. Biophysical Journal , (2003). 5/10/2019 Yang Yang, Candidacy Seminar

Conditions in Two-State Receptor Model Receptor autophosphorylation rates are proportional to the receptor activity: Only the inactive or active receptors can be methylated or demethylated. The association rates between receptors and CheR or CheBp are linearly related to the receptor activity, whiledissociation rates are independent with 𝜆. Then the inverse of the methylation or demethylation MM constants are linearly related to the receptor activity: The ratios between methylation catalytic rates and demethylation catalytic rates for the next methylation level are constant: The phosphate transfer rates from CheA to CheB or CheY are proportional to receptor activities: 5/10/2019 Yang Yang, Candidacy Seminar

Conditions for Perfect Adaptation: Protein Concentrations

Summary of Protein Concentrations Main point is that they are model dependent, with large amount of experimental variability in their measurement where these measurements are available 5/10/2019 Yang Yang, Candidacy Seminar

Relationship Between Protein Concentrations (M) (M) (M) (M) 5/10/2019 Yang Yang, Candidacy Seminar

Relationship Between Protein Concentrations (cont’d) (M) (M) (M) (M) 5/10/2019 Yang Yang, Candidacy Seminar

Relationship between Protein Concentrations (cont’d) (M) (M) (M) (M) 5/10/2019 Yang Yang, Candidacy Seminar

Summary For more precise adaptation: CheR concentration is restricted in a narrow small-value region while total receptor and CheY concentration can vary in a wide region. CheR concentration is proportional to the CheB concentration For smaller or larger value of CheY concentration, total receptor concentration can vary in a wide region. CheB concentration is restricted in a narrow region while total receptor and CheY concentration can vary in a wide region. Concluding remarks for this part

Diversity of Chemotaxis Systems In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present. Response regulator Phosphate “sink” CheY1 CheY2 Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue. 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Two CheY System Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ: CheY1p (µM) Time(s) Time(s) Requiring: Faster phosphorylation/autodephosphorylation rates of CheY2 than CheY1 Faster phosphorylation rate of CheB 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Conclusions Successful implementation of a novel method for elucidating regions in parameter space allowing precise adaptation Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of Conditions required for perfect adaptation, consistent with and extending previous works [1-3] Numerical ranges for experimentally unknown or partially known kinetic parameters Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs [1] Barkai & Leibler, Nature (1997). [2] Yi et al., PNAS (2000). [3] Tu & Mello, Biophys. J. (2003). 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar Future Work Extension to other signaling networks vertebrate phototransduction mammalian circadian clock allowing determination of parameter dependences underlying robustness of adaptation b) plausible numerical values for unknown network parameters 5/10/2019 Yang Yang, Candidacy Seminar

Vertebrate Phototransduction cGMP: cyclic GMP PDE: cGMP phosphodiesterase GCAP: guanylyl cyclase activating, Ca2+ binding protein gc: guanylyl cyclase, which synthesis cGMP http://www.fz-juelich.de/inb/inb-1/Photoreception/ 5/10/2019 Yang Yang, Candidacy Seminar

Light Adaptation of Phototransduction An intracellular recording from a single cone stimulated with different amounts of light. Each trace represents the response to a brief flash that was varied in intensity. At the highest light levels, the response amplitude saturates. (Neuroscience, Purves et al., 2001) 5/10/2019 Yang Yang, Candidacy Seminar

Kinetic Model for Vertebrate Phototransduction Russell D. Hamer, Visual Neuroscience (2000) 5/10/2019 Yang Yang, Candidacy Seminar

Mammalian Circadian Clock From Forger et al., PNAS (2003). PERs transport CRYs to nucleus CLOCK and BMAL1 bind together CLOCK·BMAL1 binds to E box to increase Pers(Crys) transcription rates E box is the sequence CACGTG of the PER1 and CRY1 genes PERs bind with kinases CKIε/δ to be phosphorylated Phosphorylated PERs bind with CRYs Only phosphorylated PER·CRY· CKIε/δ can enter nucleus Phosphorylated PER·CRY· CKIε/δ inhibit the ability of CLOCK·BMALI to enhance transcription Increasing REV-ERBα levels repress BMAL1 transcription Activator positively regulated BMAL1 transcription http://www.umassmed.edu/neuroscience/faculty/reppert.cfm?start=0 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar 5/10/2019 Yang Yang, Candidacy Seminar

Yang Yang, Candidacy Seminar 5/10/2019 Yang Yang, Candidacy Seminar

Checking Dynamics of CheY-P with Solutions A B C D 5/10/2019 Yang Yang, Candidacy Seminar

Protein Concentration Trend Shifting 5/10/2019 Yang Yang, Candidacy Seminar

Protein Concentration Trend Shifting 5/10/2019 Yang Yang, Candidacy Seminar

Protein Concentration Trend Shifting 5/10/2019 Yang Yang, Candidacy Seminar

Protein Concentration Trend Shifting 5/10/2019 Yang Yang, Candidacy Seminar

Protein Concentration Trend Shifting 5/10/2019 Yang Yang, Candidacy Seminar

Reaction Rates Trend Shifting Protein concentrations taken from SPO’s inverse of T2 MM constant (K2R-1) inverse of T3 MM constant (K3R-1) Protein concentrations taken from Mello-Tu’s T2 autophosphorylation rate (k2a) T3 autophosphorylation rate (k3a) 5/10/2019 Yang Yang, Candidacy Seminar

Reaction Rates Trend Shifting Protein concentrations taken from SPO’s inverse of T2 MM constant (K2R-1) inverse of T3 MM constant (K3R-1) Protein concentrations taken from Mello-Tu’s T2 autophosphorylation rate (k2a) T3 autophosphorylation rate (k3a) 5/10/2019 Yang Yang, Candidacy Seminar

Reaction Rates Trend Shifting Protein concentrations taken from SPO’s inverse of T1 M-M constant (K1B-1) inverse of T2 M-M constant (K2B-1) Protein concentrations taken from Mello-Tu’s T1 autophosphorylation rate (k1a) T2 autophosphorylation rate (k2a) 5/10/2019 Yang Yang, Candidacy Seminar

Reaction Rates Trend Shifting Protein concentrations taken from SPO’s inverse of T3 M-M constant (K3B-1) inverse of T4 M-M constant (K4B-1) Protein concentrations taken from Mello-Tu’s T3 autophosphorylation rate (k3a) T4 autophosphorylation rate (k4a) 5/10/2019 Yang Yang, Candidacy Seminar

Reaction Rates Trend Shifting Protein concentrations taken from SPO’s inverse of LT1 MM constant (K1LB-1) inverse of LT2 MM constant (K2LB-1) Protein concentrations taken from Mello-Tu’s LT1 autophosphorylation rate (k1al) LT2 autophosphorylation rate (k2al) 5/10/2019 Yang Yang, Candidacy Seminar

Reaction Rates Trend Shifting Protein concentrations taken from SPO’s inverse of LT3 MM constant (K2LB-1) inverse of LT4 MM constant (K3LB-1) Protein concentrations taken from Mello-Tu’s LT3 autophosphorylation rate (k12) LT4 autophosphorylation rate (k13) 5/10/2019 Yang Yang, Candidacy Seminar

Slices of 3D Surfaces of Parameter Space 1 2 3 4 5 6 7 8 1------------------------11 9 10 11 5/10/2019 Yang Yang, Candidacy Seminar

Slices of 3D Surfaces of Parameter Space 1 2 3 4 5 6 7 8 1---------------------11 9 10 11 5/10/2019 Yang Yang, Candidacy Seminar