Bistability in ODE and algebraic models Matthew Macauley Department - PowerPoint PPT Presentation

Bistability in ODE and algebraic models Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Algebraic Biology M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 1 / 28

Bistability A system is bistable if it has two stable steady-states separated by an unstable state. ` ˘` ˘ From Wikipedia. The threshold ODE : y 1 “ ´ ry 1 ´ y 1 ´ y . M T In the threshold model for population growth, there are three steady-states, 0 ă T ă M : M “ carrying capacity (stable), T “ extinction threshold (unstable), 0 “ extinct (stable). M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 2 / 28

Types of bistability The lac operon exhibits bistability. The expression level of the lac operon genes are either almost zero (“basal levels”), or very high (thousands of times higher). There’s no “inbetween” state. The exact level depends on the concentration of intracellular lactose. Let’s denote this parameter by p. Now, let’s “tune” this parameter. The result might look like the graph on the left. This is reversible bistability. In other situations, it may be irreversible (at right). M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 3 / 28

Hysteresis For reversible bistability, the up-threshold L 2 of p is higher than the down-threshold L 1 of p . This is hysteresis: a dependence of a state on its current state and past state. Thermostat example Consider a home thermostat. If the temperature is T ă 18 (e.g., in winter), the heat is on. If the temperature is T ą 23 (e.g., in summer), the AC is on. If 18 ă T ă 23, then we don’t know whether it’s spring or autumn. M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 4 / 28

Hysteresis and the lac operon If lactose levels are medium, then the state of the operon depends on whether or not a cell was grown in a lactose-rich environment. Lac operon example Let r L s “ concentration of intracellular lactose. If r L s ă L 1 , the operon is OFF. If r L s ą L 2 , the operon is ON. If L 1 ă r L s ă L 2 , the operon might be ON or OFF. In the region of bistability p L 1 , L 2 q , one can find both induced and un-induced cells. M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 5 / 28

An ODE model of the lac operon The Boolean models we’ve seen are too simple to capture bistability. We will derive two different ODE models of the lac operon that exhibit bistability: one with 3 variables, and another with 5 variables. These ODE models were designed using Michaelis–Menten equations from mass-action kinetics which we learned about earlier. They will also incorporate other features, such as: dilution of protein concentration due to bacterial growth degredation (decay) of protein concentration time delays After that, we’ll see how bistability can indeed be captured by a Boolean model. In general, bistable systems tend to have positive feedback loops (in their “wiring diagrams”) or double-negative feedback loops (=positive feedback). M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 6 / 28

Modeling dilution in protein concentration due to bacterial growth E. coli grows fast! It can double in 20 minutes. Thus, ODE models involving concentration can’t assume volume is constant. Let’s define: V “ average volume of an E. coli cell. x “ number of molecules of protein X in that cell. Assumptions: dV cell volume increases exponentially in time: dt “ µ V . dx degradation of X is exponential: dt “ ´ β x . The concentration of X is r x s “ x V . The derivative of this is (by the quotient rule): ` ˘ 1 ` ˘ 1 ` ˘ x d r x s x 1 V ´ V 1 x “ V 2 “ ´ β xV ´ µ Vx V 2 “ ´ β ` µ V “ ´p β ` µ qr x s . dt M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 7 / 28

Modeling of lactose repressor dynamics Assumptions Lac repressor protein is produced at a constant rate. Laws of mass-action kinetics. Repressor binds to allolactose : d r RA n s K 1 “ K 1 r R sr A s n ´ r RA n s Ý Ý á R ` nA RA n â Ý Ý dt 1 d r RA n s r RA n s Assume the reaction is at equilibrium: “ 0, and so K 1 “ r R sr A s n . dt The repressor protein binds to the operator region if there is no allolactose : d r OR s K 2 Ý Ý á O ` R OR “ K 2 r O sr R s ´ r OR s . â Ý Ý dt 1 d r OR s r OR s Assume the reaction is at equilibrium: “ 0, and so K 2 “ r O sr R s . dt M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 8 / 28

Modeling of lactose repressor dynamics r OR s Let O tot “ total operator concentration (a constant). Then, using K 2 “ r O sr R s , O tot “ r O s ` r OR s “ r O s ` K 2 r O sr R s “ r O sp 1 ` K 2 r R sq . r O s 1 Therefore, O tot “ 1 ` K 2 r R s . “ Proportion of free (unbounded) operator sites. ” Let R tot be total concentration of the repressor protein (constant): R tot “ r R s ` r OR s ` r RA n s � ( Assume only a few molecules of operator sites per cell: r OR s ! max r R s , r RA n s : R tot « r R s ` r RA n s “ r R s ` K 1 r R sr A s n R tot Eliminating r RA n s , we get r R s “ 1 ` K 1 r A s n . Now, the proportion of free operator sites is: ¨ 1 ` K 1 r A s n 1 ` K 1 r A s n “ 1 ` K 1 r A s n r O s 1 1 “ 1 ` K 2 r R s “ , R tot K ` K 1 r A s n O tot 1 ` K 2 p 1 ` K 1 r A s n q loooooomoooooon : “ f pr A sq where K “ 1 ` K 2 R tot . M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 9 / 28

Modeling of lactose repressor dynamics Summary The proportion of free operator sites is “ 1 ` K 1 r A s n r O s where K “ 1 ` K 2 R tot . , O tot K ` K 1 r A s n loooooomoooooon : “ f pr A sq Remarks The function f pr A sq is (almost) a Hill function of coefficient n . f pr A s “ 0 q “ 1 K ą 0 “basal level of gene expression.” f is increasing in r A s , when r A s ě 0. r A sÑ8 f pr A sq “ 1 lim “with lots of allolactose, gene expression level is max’ed.” M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 10 / 28

Modeling time-delays The production of mRNA from DNA via transcription is not instantaneous; suppose it takes time τ ą 0. Thus, the production rate of mRNA is not a function of allolactose at time t , but rather at time t ´ τ . Suppose protein P decays exponentially, and its concentration is p p t q . ż t ż t dp dp dt “ ´ µ p ù ñ p “ ´ µ dt . t ´ τ t ´ τ Integrating yields ˇ ˇ t t p p t q ˇ ˇ ln p p t q ˇ t ´ τ “ ´ µ t ˇ t ´ τ dt “ ln p p t ´ τ q “ ´ µ r t ´ p t ´ τ qs “ ´ µτ. p p t q p p t ´ τ q “ e ´ µτ , and so Exponentiating both sides yields p p t q “ e ´ µτ p p t ´ τ q . M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 11 / 28

A 3-variable ODE model of the lac operon Consider the following 3 quantities, which represent concentrations of: M p t q “ mRNA, B p t q “ β -galactosidase, A p t q “ allolactose. Assumption : Internal lactose ( L ) is available and is a parameter. The model (Yildirim and Mackey, 2004) 1 ` K 1 p e ´ µτ M A τ M q n M dM dt “ α M K ` K 1 p e ´ µτ M A τ M q n ´ r γ M M dB A L dt “ α B e ´ µτ B M τ B ´ r γ B B dA L A B dt “ α A B K L ` L ´ β A B K A ` A ´ r γ A A These are delay differential equations , with discrete time delays due to the transcription and translation processes. X There should (?) be a self-loop at M , B , and A , but we’re omitting them for clarity. M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 12 / 28

A 3-variable ODE model of the lac operon ODE for β -galactosidase ( B ) dB dt “ α B e ´ µτ B M τ B ´ r γ B B , Justification : γ B B “ γ B B ` µ B represents loss due to β -galactosidase degredation and dilution from r bacterial growth. Production rate of β -galactosidase, is proportional to mRNA concentration. τ B “ time required for translation of β -galactosidase from mRNA, and M τ B : “ M p t ´ τ B q . e ´ µτ B M τ B accounts for the time-delay due to translation. M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 13 / 28

A 3-variable ODE model of the lac operon ODE for mRNA ( M ) 1 ` K 1 p e ´ µτ M A τ M q n dM dt “ α M K ` K 1 p e ´ µτ M A τ M q n ´ r γ M M Justification : γ M M “ γ M M ` µ M represents loss due to mRNA degredation and dilution from r bacterial growth. Production rate of mRNA [=expression level!] is proportional to the fraction of free operator sites, “ 1 ` K 1 A n r O s K ` K 1 A n “ f p A q . O tot τ M “ time required for transcription of mRNA from DNA, and A τ M : “ A p t ´ τ M q . The term e ´ µτ M A τ M accounts for the time-delay due to transcription. M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 14 / 28

A 3-variable ODE model of the lac operon ODE for allolactose ( A ) dA L A dt “ α A B K L ` L ´ β A B K A ` A ´ r γ A A Justification : γ A A “ γ A A ` µ A represents loss due to allolactose degredation and dilution from r bacterial growth. The first two terms models the chemical reaction catalyzed by the enzyme β -galactosidase: α A β A L Ý Ñ A Ý Ñ glucose ` galactose . M. Macauley (Clemson) Bistability in ODE and algebraic models Algebraic Biology 15 / 28