Goal Show the modeling process used by both Collins (toggle switch) and Elowitz (repressilator) to inform design of biological network necessary to encode desired dynamical behavior (bi-stability and oscillation, respectively).

Mathematical models predict qualitative behaviors of biological systems.. Bi-stability in genetic toggle switch [1] Oscillation in genetic oscillator [2] Reversible flipping of an integrase-driven bit [3] Counting cellular events [3] [1] Collins, Cantor 2000 [2] Elowitz 2000 [3] Us

Mathematics can predict qualitative behaviors of biological systems.. Bi-stability in genetic toggle switch [1] Oscillation in genetic oscillator [2] Reversible flipping of an integrase-driven bit [3] Counting cellular events [3] [1] Collins, Cantor 2000 [2] Elowitz 2000 [3] Us

Requirements Bi-stable [1] : holds two states Inducible switch between states [1] More than one attraction state; two stable equilibria in this case.

Design [1] [1] Two different designs. pTAK plasmids have lacI repressor (IPTG inducible) and ptrc-2 promoter pair and PLs1con promoter with a temperature-sensitive repressor (cIts). pIKE plasmids have PLtetO-1 promoter in conjunction with the Tet repressor (tetR). pTAK plasmids switched by IPTG or thermal pulse. pIKE switched by IPTG or atC. IPTG LacIRep2[1] Thermal induction or atC

State variables [1] U: repressor 1 V: repressor 2 [1] Repressor concentration are the continuous dynamical state variables

RNA polymerase binding Open-complex formation Transcript elongation Transcript termination Repressor binding Ribosome binding Polypeptide elongation The dependence of transcription rate : Cooperativity Repression Decay rates of protein, and messenger RNA Beta [1] : cooperativity in repression of promoter Alpha : is the rate of protein synthesis [1] The cooperativity arises from the multimerization of the repressor proteins and the cooperative binding of repressor multimers to multiple operator sites in the promoter. U Repressor accumulation: cooperative repression of constitutively transcribed promoter Decay: Repressor degradation /dilution Parameters ODEs Parameters simplifications Cellular events complexity

Possible outcomes No steady state, mono stable [1], bi-stable [1] One repressor always shuts down the other

Question What parameter values yield bi-stability?

Accumulation: cooperative repression of constitutively transcribed promoters Decay: Degradation/dilution of the repressors Coupled first-order ODEs

Find steady state [1] [1] Solution to both ODEs = 0, for a given set of parameter values 0 0

Solutions [1] U V V’=0 U’=0 [1] Across range of U, V given parameters : alpha=1, b=y=3 0 0

One equilibrium (intersection) point for the system U V V’=0 U’=0 0 0

Evaluate across the parameter space to find bi-stability (> 1 intersection) Alpha=1 Alpha=2 Alpha=3 Increasing cooperativity Increasing synthesis rate b=y=1 b=y=2 b=y=3 U V

Bi-stable [1] when repressor expression rate and cooperativity are high Alpha=1 b=y=1 Alpha=2 Alpha=3 b=y=2 b=y=3 [1] Multiple intersections arise from sigmoidal shape, at b, y > 1, and high rate of repressor synthesis.

Similar [1] to what Collins shows U V V’=0 U’=0 [1] Parameters : alpha=2, b=y=3

Vector field shows system will move towards steady state U V Vector field [1] Parameters : alpha=2, b=y=3

U V U=2 V=0.3 Time V’=0 (blue) U’=0 Initial condition: high U V U Repressor level Approaches one steady state if initial condition is high repressor 1 [1] Parameters : alpha=2, b=y=3

U V V’=0 (blue) U’=0 V U Alter dynamic balance with inducer, repressor 2 maximally expressed. [1] Parameters : alpha=2, b=y=3

U V V’=0 (blue) U’=0 V U New initial condition for the simulation: settles into new steady state. V=2 U=0.3 Time Initial condition: high V Repressor level [1] Parameters : alpha=2, b=y=3

Mono-stability [1] [1] Single steady-state for parameters : alpha=1, b=y=3 Initial condition: high V Initial condition: high U

To achieve bi-stability 1. Balanced and high rate of repressor synthesis 2. High co-operativity of repression 3. Induction to alters dynamic balance

Choose biological components (promoters / RBS / repressors) that meet these requirements!

Mathematics can predict qualitative behaviors of biological systems.. Bi-stability in genetic toggle switch [1] Oscillation in genetic oscillator [2] Reversible flipping of an integrase-driven bit [3] Counting cellular events [3] [1] Collins, Cantor 2000 [2] Elowitz 2000 [3] Us

Requirements Oscillation [1] No settling into steady state

Design [1]

State variables [1] 3 mRNA 3 repressor proteins [1] Repressor and mRNA concentration are the continuous dynamical state variables

Possible outcomes Steady state, or oscillation

Question What parameter values yield oscillation?

Accumulation: cooperative repression of constitutively transcribed promoters Decay: Degradation/dilution of the repressors Six coupled first-order ODEs Detailed discussion of parameters in appendix

Repressor logic embedded in equations The appropriate protein represses the appropriate mRNA synthesis and translation

Predict system behavior with respect to ODE parameters Linear algebra Prediction of parameter values that yield steady state and oscillation

Dynamic stability region with respect to parameters Unstable Stable Strength of repressors Protein / mRNA degradation 1

Target : similar protein and mRNA degradation, minimal leakage (large drop in mRNA synthesis when repressed) Unstable Stable Strength of repressors Protein / mRNA degradation 1

No leakage, high repressor expression Parameters : alpha=50, alpha0=0, beta=0.2, n=2

Leakage causes steady state Parameters : alpha=50, alpha0=1, beta=0.2, n=2

Parameters : alpha=2, alpha0=0, beta=0.2, n=2 Low repressor expression causes steady state

Process Requirements Design Model: state variables, parameters Question: the issue model needs to resolve Collins Steady state analysis Explore parameter space Simulation Elowitz Find stability region Set parameters Simulation Understand parameter settings that encode desired dynamical behavior. Choose biological parts that adhere to parameter settings.

Iterate Requirements Design Model: state variables, parameters Question: the issue model needs to resolve Collins Steady state analysis Explore parameter space Simulation Elowitz Find stability region Set parameters Simulation Understand parameter settings that encode desired dynamical behavior. Choose biological parts that adhere to parameter settings.

Appendix

RNA polymerase binding Open-complex formation Transcript elongation Transcript termination Repressor binding The dependence of transcription rate : Cooperativity Repression Decay rates of protein, and messenger RNA : cooperativity of repression of promoter : synthesis : leaky [2] synthesis [1] Number of protein copies per cell n the presence of saturating amounts of repressor (owing to the `leakiness' of the promoter) [2] Here we consider only the symmetrical case in which all three repressors are identical except for their DNA-binding specificities. [3] Time is rescaled in units of the mRNA lifetime [4] mRNA concentrations are rescaled by their translation efficiency, the average number of proteins produced per mRNA molecule. U mRNA accumulation: cooperative repression of mRNA synthesis [5] Decay: Repressor degradation /dilution Parameters : mRNA model ODEs continuous dynamical state variables, (repressor concentration) Parameters simplifications Cellular events complexity

Ribosome binding Polypeptide elongation : protein to mRNA decay rate ratio [1] [1] Time is rescaled in units of the mRNA lifetime [2] Protein concentrations are written in units of KM, the number of repressors necessary to half-maximally repress a promoter; Repressor accumulation: cooperative repression of proteins produced by mRNA [5] Decay Parameters : Repressor protein model ODEs continuous dynamical state variables, (repressor concentration) Parameters simplifications Cellular events complexity Degradation and dilution : protein to mRNA ratio [1]