Bistability and a differential equation model of the lac operon Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4500, Spring 2016 M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 1 / 23
Bistability A system is bistable if it is capable of resting in two stable steady-states separated by an unstable state. � ✟� ✟ The threshold ODE : y ✶ ✏ ✁ ry From Wikipedia. 1 ✁ y 1 ✁ y . M T In the threshold model for population growth, there are three steady-states, 0 ➔ T ➔ M : M ✏ carrying capacity (stable), T ✏ extinction threshold (unstable), 0 ✏ extinct (stable). M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 2 / 23
Types of bistability For an example of bistability, consider the lac operon. 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 precise expression level depends on the concentration level 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 & an ODE model of the lac operon Math 4500, Spring 2016 3 / 23
Hysteresis In the case of reversible bistability, note that 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 set for 72 ✆ . If the temperature is T ➔ 71, then the heat kicks on. If the temperature is T → 73, then the AC kicks on. If 71 ➔ T ➔ 73, then we don’t know whether the heat or AC was on last. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 4 / 23
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 denote the concentration of intracellular lactose. If r L s ➔ L 1 , then the operon is OFF. If r L s → L 2 , then the operon is ON. If L 1 ➔ r L s ➔ L 2 , then the operon could be ON or OFF. The region of bistability ♣ L 1 , L 2 q has both induced and un-induced cells. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 5 / 23
Hysteresis and the lac operon The Boolean network models we’ve seen are too simple to capture bistability. We’ll see two different ODE models of the lac operon that exhibit bistability. These ODE models were designed using Michaelis–Menten equations from mass-action kinetics which we learned about earlier. In a later lecture, we’ll see how bistability can indeed be captured in a Boolean network system. 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 & an ODE model of the lac operon Math 4500, Spring 2016 6 / 23
Modeling dilution in protein concentration due to bacterial growth E. coli grows fast! It can double in 20 minutes. Thus, reasonable ODE models involving concentration shouldn’t assume that volume is constant. Let’s define: V ✏ average volume of an E. coli bacterial cell. Let x ✏ number of molecules of protein X in that cell. Assumptions about these derivatives: 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 , and the derivative of this is (by the quotient rule): � ✟ 1 � ✟ 1 � ✟ x d r x s x ✶ V ✁ V ✶ x ✏ V 2 ✏ ✁ β xV ✁ µ Vx V 2 ✏ ✁ β � µ V ✏ ✁♣ β � µ qr x s . dt M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 7 / 23
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 á R � nA 1 RA n â Ý dt 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 O � R Ý á 1 OR ✏ K 2 r O sr R s ✁ r OR s . â Ý dt 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 & an ODE model of the lac operon Math 4500, Spring 2016 8 / 23
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 s♣ 1 � K 2 qr R s . 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 q ☎ 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 O tot ✏ 1 � K 2 r R s ✏ , R tot K � K 1 r A s n 1 � K 2 ♣ ❧♦♦♦♦♦♦♠♦♦♦♦♦♦♥ : ✏ f ♣r A sq where K ✏ 1 � K 2 R tot . M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 9 / 23
Modeling of lactose repressor dynamics Summary The proportion of free operator sites is O tot ✏ 1 � K 1 r A s n r O s where K ✏ 1 � K 2 R tot . , K � K 1 r A s n ❧♦♦♦♦♦♦♠♦♦♦♦♦♦♥ : ✏ f ♣r A sq Remarks The function f ♣r A sq is (almost) a Hill function of coefficient n . f ♣r A s ✏ 0 q ✏ 1 K → 0 “minimal basal level of gene expression.” f is increasing in r A s , when r A s ➙ 0. r A sÑ✽ f ♣r A sq ✏ 1 lim “with lots of allolactose, gene expression level is max’ed.” M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 10 / 23
Modeling time-delays The production of mRNA from DNA via transcription is not an instantaneous process; suppose it takes time τ → 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 ♣ t q . ➺ t ➺ t dp dp dt ✏ ✁ µ p ù ñ p ✏ ✁ µ dt . t ✁ τ t ✁ τ Integrating yields ✞ ✞ p ♣ t q ✞ t ✞ t ln p ♣ t q t ✁ τ ✏ ✁ µ t t ✁ τ dt ✏ ln p ♣ t ✁ τ q ✏ ✁ µ r t ✁ ♣ t ✁ τ qs ✏ ✁ µτ. ✞ ✞ p ♣ t ✁ τ q ✏ e ✁ µτ ,and so p ♣ t q Exponentiating both sides yields p ♣ t q ✏ e ✁ µτ p ♣ t ✁ τ q . M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 11 / 23
A 3-variable ODE model Consider the following 3 quantities, which represent concentrations of: M ♣ t q ✏ mRNA, B ♣ t q ✏ β -galactosidase, A ♣ t q ✏ allolactose. Assumption : Internal lactose ( L ) is available and is a parameter. The model (Yildirim and Mackey, 2004) dt ✏ α M 1 � K 1 ♣ e ✁ µτ M A τ M q n dM K � K 1 ♣ e ✁ µτ M A τ M q n ✁ r γ M M dB dt ✏ α B e ✁ µτ B M τ B ✁ r γ B B dA L A 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. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 12 / 23
3-variable ODE model ODE for β -galactosidase ( B ) dB dt ✏ α B e ✁ µτ B M τ B ✁ r γ B B , Justification : r γ B B ✏ γ B B � µ B represents loss due to β -galactosidase degredation and dilution from 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 ♣ t ✁ τ B q . e ✁ µτ B M τ B accounts for the time-delay due to translation. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 13 / 23
3-variable ODE model ODE for mRNA ( M ) dt ✏ α M 1 � K 1 ♣ e µτ M A τ M q n dM K � K 1 ♣ e ✁ µτ M A τ M q n ✁ r γ M M Justification : r γ M M ✏ γ M M � µ M represents loss due to mRNA degredation and dilution from bacterial growth. Production rate of mRNA is proportional to fraction of free operator sites, O tot ✏ 1 � K 1 r A s n r O s 1 � K 1 r A s n ✏ f ♣r A sq . The constant τ M → 0 represents the time-delay due to transcription of mRNA from DNA. Define A τ M : ✏ A ♣ t ✁ τ M q . The term e ✁ µτ M A τ M accounts for the concentration of A at time t ✁ τ M , and dilution due to bacterial growth. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Spring 2016 14 / 23
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