Presentation on theme: "A cell constitutes the following"— Presentation transcript:
1 A cell constitutes the following External signalcytoplasmsubstrateCell wall +MembraneProducttranslationProteinSecondary messengeractivationnucleusTranslocationTFActive TFmRNAtranscriptionDNACell is dynamic as it involves in various processes;(a) it change its shape, i.e., it is elastic(b) it can communicate with other cells (cell-to-cell communication)(c) it can move (molecular motors)(d) It can process signals and involves in transport of molecules(e) Cell can also divide (Mitotic) and can make multiple copies (Replication)
2 transcription and translational processes ‘GENETIC’ Information is also passed on from one generation to another throughtranscription and translational processes‘ENVIRONMENTAL’ Information flows in and out the cell through protein synthesis .Simplified version of earlier cell diagram.Genetic Info.DNAmRNAProteinRegulationBy proteinsPhysiologyTranscriptionTranslationPolypeptideProteins (Inactive)Phosphorylation-dephosphorylation(P-D)acetylationActive (changes the conformation of protein)
3 Biological regulatory network (BRN) consists of sets of DNA’s, proteins and enzymes that involves in mutually regulating each other.The regulation gives rise to (a) functioning of the cell at normal as well as in adverse conditions (b) set a stage for developmental by inducing phenotypic variations.Regulation occurs at both transcription and translation.Regulation involves multiple feedback loopsFeedback loops are classified as positive and negative.Difficult to asess the complexity of feedback loops globally.So intution alone is not sufficient to understand. Requires mathematical modelling.Various tools and methodologies are available for modelling.
4 Usefulness of Mathematical models in BRN’s a) To account for experimental observations and to determine the validity of experimental conclusions.(b) Clarification of hypothesis.(c) Difficult to rely only on intuition. Mathematical equations provides a strong foundation for validating concepts and analyze complex data that involves multiple coupled variables.(d) Models identify critical parameters for which certain phenomena can occur.
5 Different regulatory mechanisms can be explored through models from which only plausible mechanisms can be identified. This cannot be carried out through experiments, which are expensive and time consuming.Models can help to identify different dynamically important regimes, which may be hard or inaccessible to experimentalists.Models can suggest experimentalists to perform new experiments to explore unknown, but biologically important and interesting regimes. Conversely, it can also validate the suitability of the models.Mathematical structure of the model helps to identify the similar regulatory processes that run through various biosystems
6 Differential equations How to model BRN’s ?Differential equationsOrdinary (ODE) Partial (PDE) stochastic (SDE) Functional (FDE)Further can be classified as Linear and non-linear differential equations.To model GRN‘s the universal law of chemical mass action kinetics is used widely.To use mass action kinetics, static biological circuit diagrams should beConstructed.To construct biological circuits, interaction among various proteins and types of interactions are necessary. This is transfomed in to mathematical equations.
7 In many systems, both positive and negative feedback loops together play an Important role.Take different types of positive and negative feedback loops and analyze for different dynamical phenomena.Visualization is important for to understand biological systems.Certain conclusion can be made by visualizing simple circuit diagrams. TheseCircuit diagrams are the backbone and basis of large circuit diagrams.After construction of biological circuit diagram, mathematical models are constructed. Models again can be visualized by two ways and arePhase plane analysis (b) bifurcation diagrams.We shall look at the first aspect now namelyImportance of feedback and inference from feedback circuit diagram andthe mathematical model based on feedback circuit diagram at a latter stage.
8 Classification of Feedback loops. Positive feedback loop brings aboutInstability (explosive growth)Amplification of a weak signal(c) Multistability (Multistationarity) and epigenetic modificationsNegative feedback loop brings aboutHomeostasisControl the unexplosive growthInduces oscillations (not in all cases)Robust again perturbations.
9 Terminologies and classification Symbol NameInduction+DNA-Proteininteraction- Repressiongeneg1g2m-RNA (Processed)ProteinP1P2+Positive regulationNegative regulationInhibitionProtein-Proteininteraction
10 - - For present representation g1 p1 A B g2 P2 g1 g2 P1 A B This circuit diagram is theabstraction of the whole network.
11 + - + - + What are the other circuits with feedback loops + Positive - negativeAutregulatory feedback loop+ABPositive feedback loopAB++-BCA+BCA-Negative feedback loop(daisy chain)+Feedback loops are determined by the parity of the negative signs
12 Two and three element feedback loops (non-homogenous or mixed feedback loops) ++ABAB++BistabilityOscillations-+AB++CNegative feedback loopPositive feedback loopOscillationsThere is a common element ‘c’ that connects these feedback loops. It is called‘Interlocked feedback loop’: Example is Circadian rhythms.
13 Some definitionsNonlinear dynamics: The rate of change of variables can be written as the linearfunction of the other related variables. Most nonlinear systemsexhibit interesting dynamics like bistability and oscillations.Fixed point: It is the point where the rate of change of all variables are exactly Zero.Small perturbations of the fixed point will bring the system back tothe same point and perturbations are quenched. It is then called stable.Multistability: Having more than one fixed point.Bistability: A system that has two stable fixed points.
14 Phase plane analysis.Its a graphical approach and can be only performed only for 2D systems.Its only a qualitative analysis.Its only for autonomous system. Where time does not occur explicitly in the differential equations. For examplexRight hand side of this equation, ‘t’ is not involved.The plot of x .vs. y gives the phase planeThe trajectories in the phase plane is called the phase portrait.To find the critical points or steady state or Equilibrium pointsyTrajectory
15 Linear Stability Analysis Example of linear system This is the assumed Eigen valueExample of linear systemEigen vectorThis is theassumedsolutioncoupledLinearEquationsMatrix form ofcoupledLinearEquationsCharacteristicEquationCharacteristicpolynomialVery importantMatrix!!!!!Eigen ValuesTo determineSteady stateOr Fixed point
16 Example: 1 Critical Point Called as STABLE NODE Equation Matrix form TrajectoriesStarting fromDifferentInitial conditionsEquilibriumPointPhase potraintEigen valuesREAL,SAME SIGNAND UNEQUALIC-1Eigen vectorsFor differentEigen valuesIC-2THE DYNAMICAL STATE ISSTABLE NODETime series
17 Example :2 Critical Point Called as SADDLE POINT Equation Matrix form EquilibriumPointPhase potraitEigen valuesREAL,OPPOSITE SIGNAND UNEQUALEigen vectorsFor differentEigen valuesTHE DYNAMICAL STATE ISSADDLE POINTTime series
18 If the real part of eigen value is positive Example :3CriticalPointCalled asStableSpiralEquationA =Matrix formEquilibriumPointPhase potraitEigen valuesREAL, andCOMPLEXTHE DYNAMICAL STATE ISSTABLE SPIRAL/ FOCUSIf the real part of eigen value is positiveThen the critical point is unstableSpiral / Focus.Time series
20 Suppose if it is nonlinear equation, how to determine various dynamical states. It is Simple: Linearize nonlinear system around steady state.Linear stability analysis around the steady states using Taylor expansion.What is Taylors expansion ?Consider a function f(x,y) and expand around steady state (x*,y*).
21 This is the final form of the linearized Nonlinear equation at the steadystateTwo things are to be determinedJacobianEigen values of the characteresticEquation.
22 Two steady statesExample 4: Lotka-volterraPrey-predator modelJacobianFixed or Equilibrium pointsforIs a saddleEigenvaluesforIs a Center/oscillatoryEigenvalues
23 Time series and phase portrait for α = 2, β =0. 002, γ = 0 Time series and phase portrait for α = 2, β =0.002, γ = and δ =2In three dimension with time as the axis
24 - NATURE OF FEEDBACK LOOPS AND DYNAMICS What dynamics is expected when Self regulatory feedback loopmutual induction / mutual repressionThree elements repressing each other in a daisy chain oractivationg each other in a daisy chainDifficult to perform experiments unless certain predictions can be madeThese predictions areWhat conditions will bring bistability ?(b) What conditions will bring about oscillations ?Use mathematical models to findout and use the guidelines toBuild the regulatory circuit.So (a) first find a naturally occuring simple circuit.(b) Modify the circuit to build the circuit that exhibit desired dynamics.+AABBCA-
25 Nomenclature that are to be known for understanding Gene regulation E.Coli, a prokaryotic system hasA single circular chromosome(b) PLASMIDS, an extrachromosomanal DNA(Usually this is manipulated to design new circuits)Genes are referred in lower case italics, while proteins are referred in Uppercase.For example,Ecoli
28 How gene expression is monitored Gene expression is monitored through tagging the expressed gene withGREEN FLOURESCENT PROTEIN (GFP)Various variants are also available, but the principle is same.For example,InducerPOGENEGFPWhen Gene expresses, then GFPalso expresses with almost samequantitative amount.(Direction of expression)FlourescentIntensityInducer
29 PARALLEL BETWEEN ENGINEERING TERMINOLOGIES Examples
30 CONSTRUCTION OF SYNTHETIC GENE CIRCUITS GUIDEDBYMATHEMATICAL MODELS
31 Naturally ocurring system Two genes cI and croAre under the controlOf promoter PR andPRM.Cro and λ-rep binds to thePromoter and regulatesEach other production.CroλHigh Low StateCro λ Lysisλ Cro LyticNatural systemExhibits bistabilityEither can be in Lytic stage or Lysogenic state.
32 Classification of positive and negative feedback loops One element autoregulatory feedback loop+ Positive- negativeAB(Exhibits stability)(Exhibits bistability)Natural systemAnalogy:Water flowing on alldirections from a tubdue to holes.Control the flow to one direction by blocking all the holes except one.Tetracycline responsivetransactivatorModified systemHigh inducer concentrationLow inducer concentrationNo regulationFeedback regulationregulatedunregulated
33 Two element feedback loop (homogeneous) GENETIC TOGGLE SWITCH+ONAB[A]bistabilityABOff+P+ x + = + (Positive feedback loop)- x - = + (Positive feedback loop)Steady states (SS- I and II) can be seen in experiements, but unstable steadystate (USS) cannot be seen in the experiments, but realized in mathematical model.ONON[A]OffOffStimulusSWITCHGRADED RESPONSEHYSTERESIS
34 Derivation of the box equation in Gardner paper. Circuit diagram Kinetic equationsReducing toDimensionless quantityFinal box equationP M1 + PM R1R1γ + p2R1γ p2P = PromoterR = repressorConservation equationsQuestionsWhen bistabilityOccurs?What should be takenCare when plasmidsAre constructed forToggle switchRate of synthesis
35 Nullcline of above equations i.e., InterpretationsLumped parameters describesRNA-binding, transcript andpolypeptide elongation,etc.Cooperativity from multimerizationOf repressor protein and and DNA andRepressor bindingNullcline of above equations i.e.,BISTABILITYIntersects at three points and isdueTo cooperative effect, i.e.,Two SS and one USSBistability is possible only when theRate of synthesis of two repressorsare balanced.If not, only mono stability is obtainedMONO STABILITY
36 Two parameter bifurcation diagram (Formation of Cusp) Large regionCUSPbifurcationRate of repressors if balanced properly, large region of bistabilityIs possibleSlopes of the bifurcation lines are determined by the cooperativityEffect; High cooperativity effect, large bistable regime and vice versaContributes to the robustness of the system.
37 Experimental observations of Synthetic toggle switch uvNaturalWhat is observed in naturalbacteriophage lambda circuitThis is the constructed synthetictoggle switch by R-DNA technologytaking into account of the theoreticalconsiderations from modelToggle switch observed in experimentsand variations is seen by monitoringGreen flourescent protein (GFP).SyntheticInducer LacI Clts GFP StateIPTG high low high ONHeat Low high low OFF(Isopropyl B-D-thiogalactopyroniside)
38 average cannot be predicted. First synthetic construct from the model and good prediction is made aboutthe rate of synthesis and repressor stength for the occurrence of toggleswitchBut predicted only average behavior of the cell and variations about theaverage cannot be predicted.For example, the time required for switching Toggle to high state for IPTGtakes 3-6 hrs for different cells. Non-deterministic effects cannot be predicted.The switching time is very slow and is not useful for practical purpose.So different toggle switch was constructed which was temperature sensitive.Theswitching time is fast and rapid.Slow inductiontimeAnd variability wwith IPTGCell to cell variation with IPTGFast induction timeWith temperature
39 - + - + - + Three element feedback loops (homogenous) A B A B C C negative feedback loopPositive feedback loop[A]Oscillations[A](USS)SS-I (Off state)PP
40 - - B C A Numerical and Experimental observation of Repressilator LeakinessFor standardparameterDecrease incooperativityDecrease inRepressioni.e., > 0
41 Prediction from the model Presence of strong promoters that binds the protein(b) High cooperative binding of the repressors increases the range ofoscillatory regime and robustness(c) The life time of the mRNA and proteins should be similarfor strong oscillationsSynthetic network that are to oscillate should be in accordanceto the above condition
42 Design of Experimental system Single cell isolated from colonyflouresenceBrigh fieldBut there is a large variation in the cell.This is due to stochastic fluctuations of the moleculesWhen simulated with stochastic model, this variation is accounted for.
43 Since this oscillator is noisy and unstable Since this oscillator is noisy and unstable. ‘Hysteretic’ based oscillatorIs proposed.Hypothetical network is constructed and the where the degradation of theProtein is controlled by a ‘slow’ subsytem.This gives ‘Relaxation oscillation’Still there is no experimental evidence.