Parameter Values

IPTG

5.4 Circuit Mathematical Model

5.4.2 Parameter Values

The base parameter setting of the model is listed in table 5.4. Several parameter values are directly taken from the literature or are derived from literature data. For other parameters where we lack quantitative information (for example, those governing gene expression process), we use educated guesses that are biologically feasible.

111 The following parameter values are adopted from literatures¹² and are properly modified in biologically plausible region to simulate our experimental findings: growth rate of the cells (k_c1 and k_c2, 0.4 hr^-1 (Top10F’ strain) and 0.8 hr^-1 (MG1655 strain)), synthesis rate of AHL (k_A, 0.1―0.6 nM mL hr^-1), degradation rate of AHL¹⁰ (d_Ae, 0.017 hr^-1 (3OC12HSL), 0.11 hr^-1 (3OC6HSL) at pH 7.0), a carrying capacity (C_m, 2*10⁹ cells mL^-1), and the concentration of AHL necessary to half-maximally activate p_luxI promoter (K₁ and K₂, 10 nM) (103). The value of β is assumed to be 1.2 in this analysis. Death rates (d_c1 and d_c2, 0.6 – 0.9 hr^-1) were determined from a decay curve of the number of live cells after induction of the killer gene (data not shown). “D” (hr^-1) is a dilution rate and

calculated with the relation

T

D = F

_where F is a fraction of dilution and T the time

interval between each dilution event: for example, D = 0.2 for 25% discrete dilution (F

= 1/4) every hour (T) (98). The bifurcation analyses were performed using XPPAUT (http://www.math.pitt.edu/~bard/xpp/xpp.html).

In the circuit diagram (figure 5.4), the p_lac-ara-1 promoter is activated upon exposure to IPTG. Subsequently, the predator killing rate (d_c1) is increased by increased ccdB expression, and the 3OC6HSL synthesis rate (k_A2) by prey is increased. To model the impact of IPTG on the circuit activation, we introduce the following functional expressions:

[ ]

[ ]

2

1 2 2

= 1.2 + 10 IPTG

0.1 + IPTG

dc × , (5.18)

112

[ ]

[ ]

2

2 2 2

= 0.2 + 2 IPTG

0.1 + IPTG

kA × . (5.19)

Table 5.1. Reactions described in the full model (1=predator and 2 = prey; the same form of kinetics is assumed for both the predator and the prey unless noted otherwise)

Reaction Rate Description

Population dynamics ci

→ ^a k c_i( _maxi−c c_i)/ _maxi Cell growth ci → d E c_{i i i} Cell death Expression of lysis genes and decay of products

→ E_i k_Ei M_Ei Lysis protein production Ei → d E_{Ei i} Lysis protein decay

→ M_E1 1

1 1

1

ME ME

k α Q +

Transcription of lysis gene in the predator

→ M_{E2 2 2 2}

2 2

1

ME ME

ME

k Q

Q α α +

Transcription of lysis gene in the prey

MEi → d_MEiM_Ei Lysis mRNA decay Production, diffusion, and decay of AHLs

→ A_i ^b ν_Ai Synthesis of AHL

Ai → d A_{Ai ai} Decay of AHL in its source cell

i ei

A →A η_i(A_i−A_ei) Diffusion of AHL from its source cell Aei→ d A_{Aei ei} Decay of AHL in the medium

ei ai

A →A η_i(A_ei−A_ai) Diffusion of AHL to its target cell Aai → d A_{Ai i} Decay of AHL in its target cell Production and decay of transcriptional regulators

Ri

→ k M_{Ri Ri} Production of regulator Ri → d R_{Ri i} Decay of regulator

MRi

→ ν_MRi Production of regulator mRNA

MRi → d_MRiM_Ri Decay of regulator mRNA

113

Activation of transcriptional regulators by AHLs

i ai i

R +A →P k A R_{Pi ai i} Binding of AHL to its cognate regulator

i i ai

P →R +A d P_{Pi i} Dissociation of AHL-regulator complex 2P_i→Q_i k P_{Qi i}² Dimerization of AHL-regulator complex

i 2 i

Q → P d Q_{Qi i} Dissociation of AHL-regulator complex dimer

a Reactants for this reaction are not specified. Similarly, when the right-hand-side of a reaction is empty (for example for all the decay reactions), products of the reaction are not specified.

b AHLs are indexed based on the their target cells: AHL1 is produced in the prey while AHL2 is produced in the predator.

114 Table 5.2. State variables of the full model

Variable Description c_i Cell density ^a E_i [lysis-protein] ^b M_Ei [lysis-mRNA]

A_i [AHL] in the source cell A_ei [AHL] in the medium A_ai [AHL] in the target cell R_i [regulator]

M_Ri [regulator mRNA]

P_i [AHL-regulator complex]

Q_i [(AHL-regulator complex)₂]

a Cell density is measured as number of cells per mL.

b The notation [X] represents the concentration of component X.

115 Table 5.3. Kinetic parameters of the full model

Parameter Description Unit

ki Specific cell growth rate constant min^-1 cmaxi Carrying capacity for cell growth ml^-1

di Cell death rate constant nM^-1min^-1

kEi Lysis protein synthesis rate constant min^-1 dEi Lysis protein decay rate constant min^-1 kMEi Maximal rate of lysis gene transciption nMmin^-1 αMEi Sensitivity of lysis gene transcription to AHL nM^-1

dMEi Lysis mRNA decay rate constant min^-1

νMRi Transcription rate for a regulator nMmin^-1 dMRi Regulator mRNA decay rate constant min^-1

kRi Regulator synthesis rate constant min^-1

dRi Regulator decay rate constant min^-1

νAi AHL synthesis rate constant nMmin^-1

ηi AHL diffusion rate constant across the cell membrane min^-1 dAi AHL intracellular decay rate constant min^-1 dAei AHL extracellular decay rate constant min^-1 kPi AHL/regulator binding rate constant nM^-1min^-1 dPi AHL/regulator dissociation rate constant min^-1 kQi AHL-regulator complex dimerization rate constant nM^-1min^-1 dQi (AHL-regulator)₂ unbinding rate constant min^-1

116 Table 5.4. Base values for the key parameters

Parameter Description Base value

k_i Specific cell growth rate constant 0.02 min^-1a c_maxi Carrying capacity for cell growth 0.05 ^b β Cooperativity of AHL effect 1.2 (101)

d_i Cell death rate constant 4 × 10^-5 (nM^-1 min^-1) (104)

c

k_Eik_MEi/d_MEi Collective synthesis rate constant for a lysis protein

20 (nM min^-1)^d

d_Ei Decay rate constant of a lysis protein 0.02 (min^-1)^e η_i Diffusion constant of AHLs 0.5 (min^-1) (105) ^c v_Ai Synthesis rate constant of AHL 50 (nM min^-1)^f α_Ei Effecting factor for AHL 0.068 (nM^-β) ^g d_Ai Decay rate constant of AHL in the cell 0.02 (min^-1) ^e d_Aei Decay rate constant of AHL in the medium 0.03 (min^-1) ^h

a Approximately 1.5 doublings per hour.

b The volume fraction of spheres tightly packed into a cubic space is about 0.5. In a liquid culture, a cell density of 10¹⁰/mL corresponds to a volume fraction of about 0.01, assuming a cell size of 10^-15 L.

117

c Estimated based on the data from literature data.

d This value corresponds to about 20 molecules/min; it is selected so that the dimensionless parameter κ_Ei = 2.

e The degradation of AHLs and proteins inside the cell is assumed to be primarily due to dilution by cell growth. Actual degradation rate constants for proteins may be slightly larger due to proteolysis. A small increase in these parameters will have only minor effects on the system dynamics.

f The value is based on a VAI synthesis rate constant of approximately 1 nM min^-1 per nM of LuxI (106), and the assumption that the intracellular concentration of an AHL synthase is 50 nM.

g This value is chosen so that the dimensionless parameter α_I = 500.

h This can be modulated by varying the medium Ph (107) or by applying enzymes (acylase or lactonase) that can degrade AHL (108, 109).