12/24/2015Yang Yang, Candidacy Seminar1 Near-Perfect Adaptation in Bacterial Chemotaxis Yang Yang and Sima Setayeshgar Department of Physics Indiana University, Bloomington, IN

E. Coli as a Model Organism 12/24/2015Yang Yang, Candidacy Seminar2 Workhorse of molecular biology, most studied cell in all science: - Small genome (~4300 genes) - Normal lack of pathogenicity - Ease of growth in lab Basis of recent developments in biotechnology and genetic engineering, including living factory for producing human medicines Basis for understanding of fundamental cellular processes: - cellular sensory systems - regulation of gene expression - cell division, etc. Dimensions: Body size: 1 μm in length 0.4 μm in radius Flagellum: 10 μm long Cell cycle: ~ 1 hour

E. coli in Motion 12/24/2015Yang Yang, Candidacy Seminar3 From Berg & Brown, Nature (1972). Physical constants: Cell speed: μm/sec Mean run time: 1 sec Mean tumble time: 0.1 sec Increasing attractants or Decreasing repellents

Bacterial Chemotaxis “Hydrogen atom” of biochemical signal transduction networks Paradigm for two-component receptor-regulated phosphorylation pathways Accessible for study by structural, biochemical and genetic approaches The chemosensory pathway in bacterial chemotaxis and propulsion system it regulates have provided an ideal system for probing the physical principles governing complex cellular signaling and response.

Chemotaxis Signal Transduction Network in E. coli 12/24/2015Yang Yang, Candidacy Seminar 5 Histidine kinase Methylesterase Couples CheA to MCPs Response regulator Methyltransferase Dephosphorylates CheY-P CheB CheA CheW CheZ CheR CheY Signal Transduction Pathway Motor Response [CheY-P] Stimulus Flagellar Bundling Motion Run Tumble

Flagellar Motor in E. coli 12/24/2015Yang Yang, Candidacy Seminar6 From R. M. Berry, Encyclopedia of Life Science (2001). From P. Cluzel, et al., Science (2000).

Response to Step Stimulus 12/24/2015Yang Yang, Candidacy Seminar7 Fast responseSlow adaptation From Block et al., Cell (1982). From Sourjik et al., PNAS (2002). Steady state [CheY-P] ( and running bias) independent of value constant external stimulus (adaptation) CheY-P response: Flagellar response:

Excitation and Adaptation 12/24/2015Yang Yang, Candidacy Seminar8

Precision of Adaptation 12/24/2015Yang Yang, Candidacy Seminar9 From Alon et al. Nature (1999). Precision of adaptation = steady state tumbling frequency of unstimulated cells / steady state tumbling frequency of stimulated cells Squares: Unstimulated cells Circles: Cells stimulated at t=0 (Each point represents data from 10s motion of cells.)

Robustness of Perfect Adaptation 12/24/2015Yang Yang, Candidacy Seminar10 From Alon et al. Nature (1999). Precision of adaptation robust to 50-fold change in CheR expression … …while … Adaptation time and steady state tumbling frequency vary significantly. Robustness of perfect adaptation: precision of adaptation insenstive to network parameters

This Work: Outline 12/24/2015Yang Yang, Candidacy Seminar11  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

E. coli Chemotaxis Signaling Network 12/24/2015Yang Yang, Candidacy Seminar12  Ligand binding  Methylation  Phosphorylation phosphorylation methylation Ligand binding E=F(free form), R(coupling with CheR), B(coupling with CheB p ) E’=F(free form), R(coupling with CheR)  =o(ligand occupied), v(ligand vacuum)  =u(unphosphorylated), p(phosphorylated)

Enzymatic reaction: Where E is the enzyme, E 0 is the total enzyme concentration, 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: Michaelis-Menten Kinetics 12/24/2015Yang Yang, Candidacy Seminar13 where K m is the Michaelis-Menten (MM) constant

Reaction Rates 12/24/2015Yang Yang, Candidacy Seminar14

Approach … 12/24/2015Yang Yang, Candidacy Seminar15  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

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: Augmented System 12/24/2015Yang Yang, Candidacy Seminar16 Discretize s in range {s low, s high }

Physical Interpretation of Parameter, : Near-Perfect Adaptation 12/24/2015Yang Yang, Candidacy Seminar17  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)

 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: Implementation 12/24/2015Yang Yang, Candidacy Seminar18

Convergence from Guess to Solution 12/24/2015Yang Yang, Candidacy Seminar19 A B Starting from initial guess A, solution B is generated. By comprehensively sampling space of parameters with initial guesses, solution “surfaces” are constructed. T 3 autophosphorylation rate (k 3a ) Inverse of T 3 MM constant (K 3R -1 ) ● 3%<  <5% ● 1%<  <3% ● 0%<  <1%

Parameter Surfaces 12/24/2015Yang Yang, Candidacy Seminar20 ● 1%<  <3% ● 0%<  <1% Surface 2D projections Inverse of T 1 methylation MM constant (K 1R -1 ) Inverse of T 1 demethylation MM constant (K 1B -1 ) T 1 autophosphorylation rate k 1a Inverse of T 1 methylation MM constant (K 1R -1 )

Slices of 3D Surfaces of Parameter Space 12/24/2015Yang Yang, Candidacy Seminar denote slices perpendicular to K 1B -1

Validation 12/24/2015Yang Yang, Candidacy Seminar22 Time (s) Concentration (µM) Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:

Violating and Restoring Perfect Adaptation 12/24/2015 Yang Yang, Candidacy Seminar 23 Step stimulus from 0 to 1e-3M at t=500s (5e+6,10) (1e+6,10) T 3 autophosphorylation rate (k 3a ) CheYp Concentration (µM) Inverse of T 3 MM constant (K 3R -1 ) Time (s)

Conditions for Perfect Adaptation: Kinetic Parameters 12/24/201524Yang Yang, Candidacy Seminar

Inverse of Methylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar25 T 0 autophosphorylation rate (k 0a ) Inverse of T 0 MM constant (K 0R -1 ) T 1 autophosphorylation rate (k 1a ) Inverse of T 1 MM constant (K 1R -1 )

Inverse of Methylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar26 T 2 autophosphorylation rate (k 2a ) T 3 autophosphorylation rate (k 3a ) Inverse of T 2 MM constant (K 2R -1 ) Inverse of T 3 MM constant (K 3R -1 )

Inverse of Methylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar27 LT 0 autophosphorylation rate (k 0al ) LT 1 autophosphorylation rate (k 1al ) Inverse of LT 0 MM constant (K 0LR -1 ) Inverse of LT 1 MM constant (K 1LR -1 )

Inverse of Methylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar28 LT 2 autophosphorylation rate (k 2al ) LT 3 autophosphorylation rate (k 3al ) Inverse of LT 2 MM constant (K 2LR -1 ) Inverse of LT 3 MM constant (K 3LR -1 )

Inverse of Demethylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar29 T 1 autophosphorylation rate (k 1a ) T 2 autophosphorylation rate (k 2a ) Inverse of T 1 MM constant (K 1B -1 ) Inverse of T 2 MM constant (K 2B -1 )

Inverse of Demethylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar30 T 3 autophosphorylation rate (k 3a ) T 4 autophosphorylation rate (k 4a ) Inverse of T 3 MM constant (K 3B -1 ) Inverse of T 4 MIM constant (K 4B -1 )

Inverse of Demethylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar31 LT 1 autophosphorylation rate (k 1al ) LT 2 autophosphorylation rate (k 2al ) Inverse of LT 1 MM constant (K 1LB -1 ) Inverse of LT 2 MM constant (K 2LB -1 )

Inverse of Demethylation MM Constant Autophosphorylation Rate 12/24/2015Yang Yang, Candidacy Seminar32 LT 3 autophosphorylation rate (k 3al ) LT 4 autophosphorylation rate (k 4al ) Inverse of LT 3 MM constant (K 2LB -1 ) Inverse of LT 4 MM constant (K 3LB -1 )

Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant 12/24/2015Yang Yang, Candidacy Seminar33 T 1 demethylation catalytic rate k 1b T 0 methylation catalytic rate k 0c T 2 demethylation catalytic rate k 2b T 1 methylation catalytic rate k 1c

Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant 12/24/2015Yang Yang, Candidacy Seminar34 T 3 demethylation catalytic rate k 3b T 2 methylation catalytic rate k 2c T 4 demethylation catalytic rate k 4b T 3 methylation catalytic rate k 3c

Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant 12/24/2015Yang Yang, Candidacy Seminar35 LT 1 demethylation catalytic rate k 1bl LT 0 methylation catalytic rate k 0cl LT 2 demethylation catalytic rate k 2bl LT 1 methylation catalytic rate k 1cl

Methylation Catalytic Rate/ Demethylation Catlytic Rate = Constant 12/24/2015Yang Yang, Candidacy Seminar36 LT 3 demethylation catalytic rate k 3bl LT 2 demethylation catalytic rate k 2cl LT 4 demethylation catalytic rate k 4bl LT 3 demethylation catalytic rate k 3cl

Summary: Reaction Kinetics 12/24/2015Yang Yang, Candidacy Seminar37 These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state (activity-based) receptor model *.  Inverse of methylation MM constants linearly decreases with autophosphorylation rates  Inverse of demethylation MM constants linearly increases with autophosphorylation rates  Ratio of methylation catalytic rates and demethylation catalytic rates for the next methylation level is constant for all methylation states * B. Mello et al. Biophysical Journal (2003).

Conditions for Perfect Adaptation: Protein Concentrations

Intrinsic Variability in Total Protein Concentrations 12/24/2015Yang Yang, Candidacy Seminar39 Total chemotaxis protein concentrations vary across clonal population of cells, as well as across cell cycles due to fluctuations in gene expression and partitioning of proteins at cell division.

Relationship Between Protein Concentrations 12/24/2015Yang Yang, Candidacy Seminar40 (M)

Relationship Between Protein Concentrations (cont’d) 12/24/2015Yang Yang, Candidacy Seminar41 (M)

Relationship between Protein Concentrations (cont’d) 12/24/2015Yang Yang, Candidacy Seminar42 (M)

Summary: Protein Concentrations 12/24/2015Yang Yang, Candidacy Seminar43 Preliminary observations : Total receptor concentration and CheY concentration have a lower and upper boundary respectly. CheR concentration is proportional to the CheB concentration CheR, CheB restricted ranges consistent with experiment *. * Li and Hazelbauer, Journal of Bacteriology, (2004).

Diversity of Chemotaxis Systems 12/24/2015Yang Yang, Candidacy Seminar44 Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue. In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present. Response regulator Phosphate “sink” CheY1 CheY2

Example: Two CheY System 12/24/2015Yang Yang, Candidacy Seminar45 Exact adaptation in modified chemotaxis network with CheY 1, CheY 2 and no CheZ: CheY1 p (µM) Time(s) Requiring:  Faster phosphorylation/autodephosphorylation rates of CheY 2 than CheY 1  Faster phosphorylation rate of CheB

Conclusions 12/24/2015Yang Yang, Candidacy Seminar46 I.Successful implementation of a novel method for elucidating regions in parameter space allowing precise adaptation II.Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of i.Conditions required for perfect adaptation, consistent with and extending previous works [1-3] ii.Numerical ranges for experimentally unknown or partially known kinetic parameters iii.Preliminary results on restrictions of total protein concentrations I.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).

Future Extensions 12/24/2015Yang Yang, Candidacy Seminar47 Extension to other signaling networks - vertebrate phototransduction [1] - mammalian circadian clock [2] allowing determination of a) parameter dependences underlying robustness of adaptation b) plausible numerical values for unknown network parameters [1] R. D. Hamer, et al., Vis Neurosci 22, (2005). [2] D. B. Forger and C. S. Peskin, PNAS 102, (2005).

Vertebrate Phototransduction 12/24/2015Yang Yang, Candidacy Seminar48 From cGMP: cyclic GMP PDE: cGMP phosphodiesterase GCAP: guanylyl cyclase gc: guanylyl cyclase

Light Adaptation in Phototransduction 12/24/2015Yang Yang, Candidacy Seminar49 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 (From Neuroscience Purves et al., 2001)

Kinetic Model for Vertebrate Phototransduction 12/24/2015Yang Yang, Candidacy Seminar50 Russell D. Hamer, Visual Neuroscience (2000)

Mammalian Circadian Clock 12/24/2015Yang Yang, Candidacy Seminar51  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-ERB α levels repress BMAL1 transcription  Activator positively regulated BMAL1 transcription From Forger et al., PNAS (2003).

Conditions in Two-State Receptor Model 12/24/2015Yang Yang, Candidacy Seminar52  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 CheB p are linearly related to the receptor activity, while dissociation rates are independent of the activity (hence, 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 phospho-transfer rates from CheA to CheB or CheY are proportional to receptor activities:

12/24/2015Yang Yang, Candidacy Seminar53

12/24/2015Yang Yang, Candidacy Seminar54

12/24/2015Yang Yang, Candidacy Seminar55

12/24/2015Yang Yang, Candidacy Seminar56

12/24/2015Yang Yang, Candidacy Seminar57

12/24/2015Yang Yang, Candidacy Seminar58

Checking Dynamics of CheY-P with Solutions 12/24/2015Yang Yang, Candidacy Seminar59 A B C D

Protein Concentration Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar60

Protein Concentration Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar61

Protein Concentration Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar62

Protein Concentration Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar63

Protein Concentration Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar64

Reaction Rates Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar65 T 2 autophosphorylation rate (k 2a ) T 3 autophosphorylation rate (k 3a ) inverse of T 2 MM constant (K 2R -1 ) inverse of T 3 MM constant (K 3R -1 ) Protein concentrations taken from SPO’s Protein concentrations taken from Mello-Tu’s

Reaction Rates Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar66 T 2 autophosphorylation rate (k 2a ) T 3 autophosphorylation rate (k 3a ) inverse of T 2 MM constant (K 2R -1 ) inverse of T 3 MM constant (K 3R -1 ) Protein concentrations taken from SPO’s Protein concentrations taken from Mello-Tu’s

Reaction Rates Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar67 T 1 autophosphorylation rate (k 1a ) T 2 autophosphorylation rate (k 2a ) inverse of T 1 M-M constant (K 1B -1 ) inverse of T 2 M-M constant (K 2B -1 ) Protein concentrations taken from SPO’s Protein concentrations taken from Mello-Tu’s

Reaction Rates Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar68 T 3 autophosphorylation rate (k 3a ) T 4 autophosphorylation rate (k 4a ) inverse of T 3 M-M constant (K 3B -1 ) inverse of T 4 M-M constant (K 4B -1 ) Protein concentrations taken from SPO’s Protein concentrations taken from Mello-Tu’s

Reaction Rates Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar69 LT 1 autophosphorylation rate (k 1al ) LT 2 autophosphorylation rate (k 2al ) inverse of LT 1 MM constant (K 1LB -1 ) inverse of LT 2 MM constant (K 2LB -1 ) Protein concentrations taken from SPO’s Protein concentrations taken from Mello-Tu’s

Reaction Rates Trend Shifting 12/24/2015Yang Yang, Candidacy Seminar70 LT 3 autophosphorylation rate (k 12 ) LT 4 autophosphorylation rate (k 13 ) inverse of LT 3 MM constant (K 2LB -1 ) inverse of LT 4 MM constant (K 3LB -1 ) Protein concentrations taken from SPO’s Protein concentrations taken from Mello-Tu’s

Slices of 3D Surfaces of Parameter Space 12/24/2015Yang Yang, Candidacy Seminar

Slices of 3D Surfaces of Parameter Space 12/24/2015Yang Yang, Candidacy Seminar

Slices of 3D Surfaces of Parameter Space Comparing Pair-Wise Relationship 12/24/2015Yang Yang, Candidacy Seminar73 T 1 autophosphorylation rate (k 1a ) Inverse of T 1 MM constant (K1 R -1 ) T 1 autophosphorylation rate (k 1a ) Inverse of T 1 MM constant (K 1B -1 )

E. coli and Bacteria Chemotaxis 12/24/2015Yang Yang, Candidacy Seminar74 Increasing attractants or Decreasing repellents