Bistability and a differential equation model of the lac operon

Transcription

1 Bistability and a differential equation model of the lac operon Matthew Macauley Department of Mathematical Sciences Clemson University Math 4500, Fall 2016 M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

2 Bistability A system is bistable if it is capable of resting in two stable steady-states separated by an unstable state. From Wikipedia. The threshold ODE: y 1 ry 1 y M 1 y 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, Fall / 23

3 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, Fall / 23

4 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, Fall / 23

5 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 rls denote the concentration of intracellular lactose. If rls L 1, then the operon is OFF. If rls L 2, then the operon is ON. If L 1 rls L 2, then the operon could be ON or OFF. The region of bistability pl 1, L 2q has both induced and un-induced cells. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

6 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, Fall / 23

7 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: cell volume increases exponentially in time: µv. dt degradation of X is exponential: βx. dx dt The concentration of x is rxs x, and the derivative of this is (by the quotient rule): V dv drxs dt x 1 V V 1 x 1 V 2 βxv µvx 1 V 2 β µ x V pβ µqrxs. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

8 Modeling of lactose repressor dynamics Assumptions Lac repressor protein is produced at a constant rate. Laws of mass-action kinetics. Repressor binds to allolactose: R na K 1 drra ns ÝáâÝ RA n K 1rRsrAs n rra ns 1 dt Assume the reaction is at equilibrium: drra ns dt 0, and so K 1 rrans rrsras n. The repressor protein binds to the operator region if there is no allolactose: O R K 2 ÝáâÝ 1 OR drors dt K 2rOsrRs rors. Assume the reaction is at equilibrium: drors dt 0, and so K 2 rors rosrrs. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

9 Modeling of lactose repressor dynamics Let O tot total operator concentration (a constant). Then, using K 2 rors rosrrs, Therefore, O tot ros rors ros K 2rOsrRs rosp1 K 2qrRs. ros 1 O tot 1 K 2 rrs. Proportion of free (unbounded) operator sites. Let R tot be total concentration of the repressor protein (constant): R tot rrs rors rra ns Assume only a few molecules of operator sites per cell: rors! max rrs, rra ns ( : Eliminating rra ns, we get rrs R tot rrs rra ns rrs K 1rRsrAs n R tot 1 K 1rAs n. Now, the proportion of free operator sites is: ros O tot where K 1 K 2R tot. 1 1 K 1 2rRs R 1 K 2p tot 1 K 1 ras q 1 K1rAs n 1 K 1 K1rAsn, 1rAs n K K 1rAs n n loooooomoooooon :f prasq M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

10 Modeling of lactose repressor dynamics Summary The proportion of free operator sites is ros O tot loooooomoooooon 1 K1rAsn, where K 1 K 2R tot. K K 1rAs n :f prasq Remarks The function f prasq is (almost) a Hill function of coefficient n. f pras 0q 1 K 0 f is increasing in ras, when ras 0. lim f prasq 1 rasñ8 minimal basal level of gene expression. with lots of allolactose, gene expression level is max ed. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

11 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 pptq. Integrating yields ln pptq t t τ» dp t dt µp ùñ µt t t τ dt ln Exponentiating both sides yields t τ dp p µ» t t τ dt. pptq µrt pt τqs µτ. ppt τq pptq ppt τq e µτ,and so pptq e µτ ppt τq. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

12 A 3-variable ODE model Consider the following 3 quantities, which represent concentrations of: Mptq mrna, Bptq β-galactosidase, Aptq allolactose. Assumption: Internal lactose (L) is available and is a parameter. The model (Yildirim and Mackey, 2004) dm 1 α M dt K K 1pe µτ M A τm q n K 1pe µτ M AτM q rγ MM n db dt α Be µτ B M τb rγ B B da dt α L AB K L L β AB A A rγ AA These are delay differential equations, with discrete time delays due to the transcription and translation processes. K A M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

13 3-variable ODE model ODE for β-galactosidase (B) Justification: db dt α Be µτ B M τb rγ B B, rγ B B γ B B µb 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 τb : Mpt τ B q. e µτ B M τb accounts for the time-delay due to translation. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

14 3-variable ODE model ODE for mrna (M) Justification: dm dt 1 K 1pe µτ M A τm q n α M K K 1pe µτ M AτM q rγ MM n rγ M M γ M M µm represents loss due to mrna degredation and dilution from bacterial growth. Production rate of mrna is proportional to fraction of free operator sites, ros O tot 1 K1rAsn 1 K 1rAs n f prasq. The constant τ M 0 represents the time-delay due to transcription of mrna from DNA. Define A τm : Apt τ M q. The term e µτ M A τm 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, Fall / 23

15 3-variable ODE model ODE for allolactose (A) Justification: da dt α AB K L L L β AB K A A A rγ AA rγ A A γ A A µa represents loss due to allolactose degredation and dilution from bacterial growth. The first term models production of allolactose from the chemical reaction lac β gal ÝÑ allo. The second term models loss of allolactose from the chemical reaction allo β gal ÝÑ glucose & galactose. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

16 A 3-variable ODE model Steady-state analysis To find the steady states, we must solve the nonlinear system of equations: 1 K 1pe µτ M A τm q n 0 α M K K 1pe µτ M AτM q rγ MM n 0 α B e µτ B M τb rγ B B L 0 α A B K L L β AB K A A A rγ AA This was done by Yildirim et al. (2004). They set L mm, which was in the bistable range. They also estimated the parameters through an extensive literature search. Finally, they estimated µ min 1 by fitting the ODE models to experimental data. Steady states A (mm) M (mm) B (mm) I II III M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

17 3-variable ODE model Figure: Bistability is pl, A q space. The y-axis is in logarithmic scale. For a range of L concentrations there are three coexisting steady states for the allolactose concentration. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

18 3-variable ODE model Figure: Time series simulations of mrna, β-galactosidase and allolactose concentrations. These were produced by numerically solving the 3-variable model using L mm, which is in the bistable region. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

19 5-variable ODE model Consider the following 5 variables, which represent concentrations of: Mptq mrna, Bptq β-galactosidase, Aptq allolactose. Pptq lac permease. Lptq intracellular lactose. The model (Yildirim and Mackey, 2004) dm dt α M 1 K 1pe µτ M A τm q n K K 1pe µτ M AτM q n Γ0 rγ MM db dt α Be µτ B M τb rγ B B da dt α L AB K L L β A AB K A A rγ AA dp dt α Pe µpτ B τ P q MτB τ P rγ P P dl dt α L e L LP β Le P K Le L e K Le L α AB K L L L rγ LL M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

20 5-variable ODE model Remarks The only difference in the ODE for M is the extra term Γ 0 which describes the basal transcription rate (in the absence of extracellular lactose). The ODEs for B and A are the same as in the 3-variable model. The ODE for P is very similar to the one for B: production rate of lac permease is proportional to mrna concentration, with a time-delay. the 2nd term accounts for loss due to degredation and dilution. The ODE for lactose, dl dt α LP K Le L e L e β Le P K L1 L L α L AB K L L rγ LL, is justified by the following: The 1st term models gain due to transport of external lactose by lac permease. The 2nd term accounts for loss due to this process being reversible. The 3rd term describes loss due to lac β gal ÝÑ allo. the 4th term accounts for loss due to degredation and dilution. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

21 A 5-variable ODE model To find the steady states, we set M 1 A 1 B 1 L 1 P 1 0 and solve the resulting nonlinear system of equations. This was done by Yildirim et al. (2004). They set L e mm, which was in the bistable range. They also estimated the parameters through an extensive literature search. Finally, they estimated µ min 1 by fitting the ODE models to experimental data. SS s A (nm) M (mm) B (mm) L (mm) P (mm) I II III M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

22 5-variable ODE model Figure: Bistability is pl, A q space. The y-axis is in logarithmic scale. For a range of L e concentrations there are three coexisting steady states for the allolactose concentration. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23

23 5-variable ODE model Figure: Time series simulations of mrna, β-galactosidase and allolactose concentrations. These were produced by numerically solving the 3-variable model using L e , which is in the bistable region. M. Macauley (Clemson) Bistability & an ODE model of the lac operon Math 4500, Fall / 23