2.18 The weak A,B mean and C,D variance errors of species A,C A and B,D C of the crystallization example, using the Direct Hybrid method, the HyJCMSS method, and the HyJCMSS method with
The Numerical Solution of Ité Stochastic Differential Equations
Definitions and Formal Solutions 000 47
Consider a system with a state vector described by N variables The state vector is
The system X = {X1, X2, , Xy} consists of real-valued variables influenced by one or more continuous-valued random processes, categorizing it as a stochastic process The Central Limit Theorem suggests that a suitable model for these continuous-valued random processes is the Wiener process, W(t) For each influencing random process, an additional Wiener process is incorporated, allowing for the mathematical modeling of diverse systems via a Wiener-driven vector stochastic differential equation (SDE) of the form dX(t) = a(X, t) dt + ∑(b_j(X, t) dW_j), where j = 1, , M represents the Wiener processes and i = 1, , N denotes the state variables The time dependence of the Wiener process is often omitted for simplicity If both a_j and b_ij are linear or constant regarding X_i, the SDE remains linear; if b_ij are constant, it exhibits additive noise, while any b_ij as a function of X_i indicates multiplicative noise This SDE notation is valuable for expressing the integral form.
The stochastic integral, represented as { f(X,t)dW, can be evaluated using either the Itô or Stratonovich definitions The classification of the stochastic differential equation (SDE) depends on the chosen definition, designating it as either an Itô or Stratonovich SDE Notably, it is possible to convert an Itô SDE into a Stratonovich SDE through a transformation of the drift coefficients.
Consider a scalar linear It6 SDE driven by a single Wiener process, dX, = (ay(t)X; +ay(t)) dt + (by (t)X, +bo(t)) dW (2.69)
It has the formal solution
X=, (x + [ (oats an(s) —bi(s)br(s)) Be! ds+ [ bá bạ(s)® 1 mi) (2.70)
CHAPTER 2 STOCHASTIC NUMERICAL METHODS 48 where ®, ,, is the fundamental solution, ®, ;, = exp ( (a6 — s8109) ds + [os av) (2.71) The mean u(t) and variance v(t) of the solution to Eq (2.70) satisfies two ordinary differential equations tH ay (ule) +an(0 (272) and
` = (2ai() +b{()) vữ) + by (1) bo (t)u(t) + bye (t) +B} (2.73)
The solution to a homogeneous linear stochastic differential equation (SDE) with specific conditions, where the functions f, by, an, and b2 are defined, results in a Gaussian distribution characterized by mean and variance as outlined in Eqs (2.72) and (2.73) In cases where these conditions are not met, the solution typically does not follow a Gaussian distribution Additionally, for a vector system comprising multiple linear SDEs influenced by various Wiener processes, applying spectral decomposition to the drift vector and diffusion tensor can transform the system into a set of decoupled linear SDEs, which are driven by a modified single Wiener process, maintaining a formal solution.
Explicit Solutions of Some Stochastic Differential Equations
A formal solution to a linear stochastic differential equation (SDE) is always available and can often be computed analytically These solutions are valuable for validating stochastic numerical integrators and assessing their errors related to time steps For instance, consider a non-homogeneous scalar linear Itô SDE with additive noise, which serves as an example for this analysis.
—X, dX, = (+ ) dt + đWUi (2.74) is satisfied by the solution t
In the second example, a homogenous scalar linear It6 SDE with multiplicative noise, or is satisfied by the solution
One effective approach to solving a non-linear system of stochastic differential equations (SDEs) is to transform it into a linear system, leveraging the existence of formal solutions for linear SDEs This can be achieved through various techniques, such as applying non-linear transformations or reconfiguring a single non-linear SDE into a system of coupled linear SDEs.
CHAPTER 2 STOCHASTIC NUMERICAL METHODS 49 scalar It6 SDE dX, = (aX!'+bX,) dt+cX, dW, (2.78) the transformation y =xˆ” 1—n converts the non-linear SDE into a linear one that satisfies t Ts 1
X, = ©; (x +a(1 -n) | er! as) (2.79) with ©, = exp ((- s2) r+eWi() (2.80)
Stochastic differential equations may also describe a complex-valued stochastic process driven by real-valued Wiener processes.
Strong and Weak Solutions 2 ee ee 49
The formal solution of a general stochastic differential equation can be approached in two ways: by generating a trajectory through individual Wiener processes, leading to a strong solution, or by determining a probability distribution that encompasses all possible trajectories, resulting in a weak solution Specifically, for the generalized vector stochastic differential equation with multiple Wiener processes, if both the drift vector and diffusion tensor are bounded and the diffusion tensor is positive definite, a weak solution exists that satisfies the Fokker-Planck equation.
THẾ) - — Yay tlhe) + YY Sep | PX) (2.81) which is a partial differential equation with N dimensions
Two different systems of stochastic differential equations can exhibit the same weak solution while possessing distinct strong solutions For instance, a two-dimensional system oscillating out of phase in a clockwise manner will yield a different strong solution compared to one oscillating counter-clockwise, despite both sharing the same weak solution Strong solutions provide more detailed information than weak solutions By generating multiple trajectories of each Wiener process and analyzing an ensemble of strong solutions, one can compute the distribution of that ensemble to derive the weak solution However, it is generally not feasible to sample a probability distribution of a time-dependent system to create a strong solution, except in specific cases.
The strong solution of a general system of stochastic differential equations (SDEs) often lacks a formal solution, necessitating the use of stochastic numerical integrators to simulate trajectories These methods produce a strong approximation to the SDE solution when the numerically generated trajectory converges to the exact trajectory, provided that the Wiener process paths are fixed and the time step approaches zero Specifically, this convergence is achieved as the time step, denoted as Δt = tj+1 - ti, decreases.
CHAPTER 2 STOCHASTIC NUMERICAL METHODS 50 and if the paths of all Wiener processes in the system at times ¢; for i= 1, m are fixed as {Wj (t1),Wj(t2), Wj(t,)} for j = 1, ,M, then the exact trajectory of a d- dimensional stochastic differential equation can be evaluated at times t; to be {X¢(t1) Xi (t2) ,Xe(tn)} for k= 1, ,d The numerical approximation of the solution of the SDE that uses the same paths of the Wiener processes is {Xx(t1) Xi(t2), ơ Ấy(„)} and is considered a strong approximation of the SDE if
A stochastic numerical method that provides a strong approximation to the solution of a stochastic differential equation will converge to the exact solution in a path-wise manner This section will concentrate on stochastic numerical methods that yield strong solutions for non-linear stochastic differential equations.
2.4.4 It6 and Stratonovich Stochastic Integrals
The Wiener process integral can be understood through various interpretations, with the Itô and Stratonovich definitions being the most significant These distinct definitions carry practical implications, which are summarized in this review.
Consider a time interval [0,7] divided into n equal partitions 0 = t < th These parameters are lumped together to create the final inducer-independent multiplicative constant © flu: Fluorescence units.
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSISOFA
The Overall Steady-State Equation
The steady-state GFP fluorescence, influenced by the concentrations of aTC and IPTG, can be represented by an algebraic equation comprising three key terms The first term, P;,;-, indicates the probability of successful Holoenzyme assembly on the promoter, calculated through thermodynamic Gibbs free energies via the chemical partition function This term is dependent on the concentrations of aTC and IPTG, as derived from the free and bound repressor concentrations The second term is a constant, o, representing the GFP fluorescence produced when the Holoenzyme is consistently assembled at the promoter (P; = 1) Lastly, the third term accounts for the background autofluorescence of cells, denoted as Cpuckground © 86 Together, these components facilitate the calculation of steady-state GFP fluorescence.
GFPss = Pinit ({aTC], (IPTG]) x Ụ + Chackground (4.4)
The concentration of the inducer specifically influences the molecular interactions involved in Holoenzyme assembly, while those interactions occurring in post-Holoenzyme assembly processes remain unaffected This distinction allows for precise fitting of experimental data to the model results.
The probability of the Holoenzyme having successfully assembled at the promoter is called the
The "transcriptionally ready state," represented as Pj,;;, is determined by assuming that protein-DNA interactions at the promoter reach chemical equilibrium We enumerate all possible regulatory states and calculate their probabilities using a canonical-like partition function The overall probability of the promoter being in this "transcriptionally ready state" is derived from the sum of regulatory states that include the Holoenzyme bound to the promoter.
The probability of the promoter existing at one of its regulatory states is calculated by using hjexp (3# )
The equation Pinit = (4.6) represents the total Gibbs free energy (AG) and the density of energy-equivalent microstates (h) for the i-th regulatory state In this context, the gas constant (R) and the physiological temperature (T) are set at 37°C.
At 310 Kelvin, the total Gibbs free energy is the aggregate of the Gibbs free energies from the molecular interactions present in that state The density of microstates, denoted as h;, can be computed using the equation h¡ = |LacL]”' [Lael:IPTGx]”° [TetR›]|”° [TetR:aTCa]”* [RNAP,]|”5 (4.7).
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSIS OF A
Table 4.4: All 55 unique regulatory states of the synthetic promoter are shown with their corre- sponding Gibbs free energies and density of microstates i | Ol O2 O3 P| aaie
5 — Tet:aTC — — Atet:uTC—DNA
7 — — Tet:aTC — A Get:uTC—DNA
8 Lac Tet — — AGiuc—pNna + AGrer—DNA
9 Lac Tet:aTC — — AGlac—pNA + AGtetuTC-DNA
10 Lac — Tet — AOT„e—DNA + A tai —DNA
Il Lac — Tet:aTC — AOwe—DNA + Â €ter:uTC—DNA
12 | LacIPTG Tet — — A laeIPTG—DNA + AGtet—DNA l3 | LacIPTG TetaTC — — A Gl„e:IPTG—DNA + A Gter:aTC—DNA
14 | LacIPTG — Tet — AGjue:IPTG-DNA + AGrer—DNA
15 | LacIPTG — Tet:aTC — A GiaeIPTG—DNA + AGtetuTC-DNA
17 — Tet:aTC Tet — AGreraTC—-DNA + AGrer—DNA
18 — Tet Tet:aTC — A Gai —DNA + A ter:uTC—DNA
19 — TetaTC Tet:aTC — 2 At:4TC—DNA
20 Lac Tet Tet — AO„e—pNA +2 AOz‡— DNA
21 Lac Tet:aTC Tet — AOize—DNA + A ai —DNA + AG tet:uTC-DNA
22 Lac Tet Tet:aTC — AOTue—DNA + Á teruTC—DNA + Â Œai—DNA
23 Lac TetaTC Tet:aTC — AOge—DNA + 2 AO(4t:uTC—DNA
24 | Lac:IPTG Tet Tet — AGtuc:IPTG—DNA + 2 AGrer—DNA
25 | LacIPTG — Tet:aTC Tet — AOwe:IPTG—DAA + Atar—DNA + Â tat:uTC—DNA
26 | Lac:IPTG Tet Tet:aTC — AGlac:IPTG-DNA + AGteruTC-DNA + AGter—DNA
27 | LacIPTG Tet:aTC Tet:aTC — AGtac:IPTG—DNA + 2 AGtet:uTC-—DNA
Holoenzyme and One Repressor Bound
29 Lac — — RNAP/ | Aize—DwA + AGRNAP-DNA
30 | Lac:IPTG — — RNAP/G | Ai¿e:IPTG—DNA + AỞRNAP—DNA
31 — Tet — RNAP/G AGrer_pNA + AGRNAP—DNA
32 — Tet:aTC — RNAP/G | AGie:uTCT—DNA + ACRNAP-DNA
33 — — Tet RNAP/G | AG/z;—DNA + AGRNAP—DNA
34 — — Tet:aTC RNAP/G | AG¡z;z7C_—DwA + AGRNAP—DNA
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSIS OFA
Table 4.5: (Continued) All 55 unique regulatory states of the synthetic promoter are shown with their corresponding Gibbs free energies and density of microstates i | ol O2 O3 Pp | agte
Holoenzyme and Two Repressors Bound
35 Lac Tet — RNAP/G A Gtuc—DNA +A Gret—DNA +A GRNAP—DNA
36 Lac Tet:aTC — RNAP/G | AGiuc—pwa + AGretuTC-DNA + AGRNAP-DNA
37 Lac — Tet RNAP/G A Gtuc—DNA +A Gret—DNA +A GRNAP_DNA
38 Lac — Tet:aTC RNAP/G | AGiz¿T—pwA + Ae:4TC—DNA + AỞRNAP—DNA
39 Lac:IPTG Tet —— RNAP/G A Glac:IPTG—DNA +A Gret—DNA +A GRNAP—DNA
40 Lac:IPTG Tet:aTC — RNAP/G A Gtac:IPTG-DNA +A GreraTC_DNA +A GRNAP—DNA
41 Lac:IPTG — Tet RNAP/G A Gtac:IPTG-DNA +A Gret—DNA +A GRNAP—DNA
42 | Lac:IPTG — Te¿aTC RNAP/G | AÓIjzejPTG-DNA + Atet:uTCT—DNA + ACRNAP—DNA
43 — Tet Tet RNAP/G | 2 AO¿;_—DxA + AGRNAP_—DNA
44 — Tet:aTC Tet RNAP/G | AGrerarc—pna + AGrer—pna + AGRNAP—DNA
45 —— Tet Tet:aTC RNAP/o AGrer-pNA + AGtet-uTC-DNA + AGRNAP-—DNA
46 — Tet:aTfC Tet:aTfTC RNAP/G | 2 AG¿:z7C_-DNA + AORNAP_—DNA
Holoenzyme and Three Repressors Bound
47 Lac Tet Tet RNAP/G | AO,e—pwA +2 AGtaT—DNA + AORNAP—DNA
48 Lac Tet:aTC Tet RNAP/o AGtae—DNA + AGrer—pNA + AGtetuTC-DNA + AGRNAP-DNA
49 Lac Tet Tet:aTC =RNAP/O | AGyuc—pna + AGreratc—DNA + AGret—pNa + AGRNAP—DNA
50 Lac Tet:aTC Tet:aTfC RNAP/G | AOze pxwA + 2 AOe:uTC—DNA + AORNAP—DNA
31 Lac:IPTG Tet Tet RNAP/o A Glac:IPTG—DNA +2 AGrer—_pna + AGRNAP-DNA
52 | Lac:IPTG Tet:aTC Tet RNAP/G | AGi¿p?gTDNA + A Gia —DAA + AigiaTCT—DNA + ARNAP-DNA
53 | Lac:IPTG Tet TetaTC RNAP/G | AGeiprơ—DNA + AŒrzzTC—DNA + Aket—DAA + Á GRNAP—DNA
34 | LacIPTG Tet:aTC Tet:aTC Repressor Non-Specific Binding RNAP/G | AGige-rprg—pna + 2 AGTet:uTC—DNA + ACRNAP—-DNA
+ The density of microstates for the non-specific binding of repressor to genomic DNA is
The regulatory expression A; = [Laca] x [Lac:IPTG4] x [TetR2] x [TetR:aTC2] represents the interaction of various species in a synthetic promoter's regulatory states, where binary variables indicate the participation of each species This formulation resembles the mass action rate law, illustrating the probability of simultaneous binding of diverse molecules at the promoter The analysis encompasses 55 distinct regulatory states, detailed in Tables 4.4 and 4.5, which account for all configurations of free and inducer-bound Lac and Tet repressors, as well as the binding states of the Holoenzyme to the promoter.
4.5 Combining Experimental and Model Results
Calculating an Unknown Parameter cuc vu 229
Using the base kinetic and thermodynamic parameters in Table 4.3, we solve Eq (4.4) over the sixteen different concentrations of aTC and IPTG inducer (from 0.001 to 2 mM IPTG and from
The steady-state average GFP fluorescence was measured at inducer concentrations ranging from 1 to 200 ng/mL (aTC), as illustrated in Figure 4.11C Initially, the model response of the synthetic promoter did not align with the experimental data, as seen in the comparison between Figures 4.11A and 4.11C However, adjustments to the parameters may be necessary to improve the model's accuracy.
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSISOFA
IPTG [mM] 00 aTC [ng/mL] IPTG [mM] aTC [ng/mL]
IPTG [mM] 00 aTC [ng/mL]
The steady-state average GFP fluorescence across sixteen concentrations of aTC and IPTG, with background autofluorescence subtracted, is analyzed alongside the steady-state mathematical model The model incorporates parameters where AGo,-rwap is +25 kcal/mol and AGo,-gxap is +0.5210 kcal/mol, while all other kinetic and thermodynamic parameters are kept constant This analysis highlights the factors listed in Table 4.3 that influence the inducer-dependent probability of achieving the "transcriptionally ready state."
The study identifies four key parameters related to the inducer-dependence of P.,, specifically AGryap—pna, AGo1—rnap, AGo2—Rnap, and AGo3_Rnap However, changes in these parameters do not align with experimental data Notably, reducing AGo_rnap to 0.52 kcal/mol allows the model to closely match experimental results, indicating that the steric interaction between Lac repressors and RNA polymerase is relatively weak, as evidenced by the data.
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSIS OF A
The L1 Norm, representing the difference between experimental data and model results, is illustrated in Figure 4.12 as a function of the thermodynamic parameter AGo,-pnap Notably, the global minimum occurs at AGo,-rnvap = +0.5210 kcal/mol.
Predicting the Behavior of Improved Synthetic Promoters 231 Discussion and Conclusions 2 Q.2 21L u g kg va 234
The synthetic promoter developed only partially meets its intended design goals, exhibiting a fuzzy AND logical gate behavior with two distinct levels of gene expression in response to aTC and IPTG inducers To enhance the promoter's AND-like response, we employ a mathematical model to predict outcomes from rearranging the genetic components within the synthetic promoter By swapping the positions of the lacO1 and tetO2 operators and recalculating the model's results, we aim to optimize the promoter while maintaining all other parameters constant.
The proposed synthetic promoters are illustrated in Figure 4.13, featuring annotated sequences that highlight our design limitations of three overlapping operators—one upstream, one downstream, and one within the spacer To effectively repress gene expression, we strategically select a repressor that binds either at the spacer or downstream positions, creating sufficient steric hindrance with RNA polymerase to inhibit transcription.
Figure 4.14 illustrates the predicted steady-state average GFP fluorescence in response to the aTC and IPTG chemical inducers for various synthetic promoters The predictions are based on the assumption that operators in the spacer and downstream positions effectively hinder RNA polymerase binding, with AGg2_pna and AGo3_pna values of +1.5 kcal/mol and +1.0 kcal/mol, respectively Evidence suggests that the middle operator in the spacer is more effective at steric repression than the downstream operator While we assume that the steric interactions of the Lac and Tet repressors with RNA polymerase are equivalent, this may not be entirely accurate Nonetheless, these assumptions can reveal general trends that may enhance the design of synthetic promoters.
The first synthetic promoter places the sole lacO1 operator in the upstream position The model
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSIS OF A
TCCCTATCAGTGATAGAGA TTGACA TCCCTATCAGTGATAGA GATACT AATTGTGAGCGGATAACAATT AGGAAACCGGTTC ATG
TCCCTATCAGTGATAGAGA TTGACA TTGTGAGCGGATAACAA GATACT TTCCCTATCAGTGATAGAGA AGGAAACCGGTTC ATG
AATTGTGAGCGGATAACAA TTGACA TIGIGAGCGGATAACAA GATACT TTCCCTATCAGTGATAGAGA AGGAAACCGGTTC ATG
-35` ——————m=m=mmm=m _ 7 () ——mmL————— | › lacO1 jacOl tetO2 RBS _
TCCCTATCAGTGATAGAGA TTGACA TTGTGAGCGGATAACAA GATACT AATTGTGAGCGGATAACAATT AGGAAACCGGTTC ATG
The proposed modifications to the synthetic promoter involve strategic placements of tetO2 and lacO1 operators to enhance gene expression responses In various configurations, such as having two tetO2 operators upstream and a lacO1 downstream, or vice versa, the resulting expression closely resembles an AND logic gate, where high gene expression occurs only in the presence of both aTC and IPTG Compared to the existing synthetic promoter, these alterations significantly improve expression levels Additionally, placing the lacO1 operator in the spacer region yields a similar response but with variations in low expression plateaus depending on the presence of inducers The positioning of the repressor is crucial for achieving a precise AND response, as the Lac repressor's weaker affinity compared to the Tet repressor can be optimized by placing lacO1 in more effective positions This strategic arrangement minimizes gene expression in the absence of either inducer, ultimately enhancing the fidelity of the AND response.
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSIS OFA
Average GFP Fluorescence Average GFP Fluorescence
IPTG [mM] 00 ATC [ng/ml] IPTG [mM] ATC [ng/ml]
Average GFP Fluorescence Average GFP Fluorescence
IPTG [mM] ATC [ng/ml] IPTG [mM] 00 ATC [ng/ml]
The predicted steady-state average GFP fluorescence of the proposed synthetic promoters, illustrated in Figure 4.13, is based on specific free energy values: AGo,~pn4 = +0.521 kcal/mol, AGo2_pna = +1.5 kcal/mol, and AGg3_pwa = +1.0 kcal/mol The first synthetic promoter exhibits the most AND-like logical response, yet it shows significant plateaus in gene expression without the presence of aTC or IPTG, due to the optimal positioning of tetO2 and lacO1 operators that enhances repression efficiency Conversely, relocating the lacO1 operator to the middle position while keeping tetO2 operators less efficiently placed leads to an increased plateau of gene expression without aTC Additionally, a single tetO2 operator in the second most efficient position also impacts expression levels, while placing a single tetO2 operator in the least efficient position markedly raises the low plateau of gene expression in the absence of aTC.
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSIS OFA
SYNTHETIC PROMOTER 234 aTC or IPTG
Using a quantitative model allows for the rapid evaluation of alternative synthetic promoters with modified DNA sequences The subsequent step involves constructing and characterizing these synthetic promoters to assess the alignment between experimental data and model predictions In cases of discrepancies, we can identify the parameters responsible for these differences and adjust their values accordingly Notably, the remaining estimated parameters, such as AGoa_-pxa and AGog3_pna, play a crucial role in explaining the variations between experimental data and model results, facilitating their calculation.
The objective was to develop a synthetic promoter that activates the expression of a gfp reporter gene only when high concentrations of both IPTG and aTC are present The initial design featured a lacO1 operator overlapping the -35 hexamer, a tetO2 operator in the spacer region, and another tetO2 operator overlapping the -10 hexamer, along with non-consensus -35 and -10 hexamers and a structured mRNA ribosome binding site These design choices aimed to minimize gene expression in the absence of the lac and tet repressors The synthetic promoter exhibited two distinct levels of gene expression in response to the inducers: a lower plateau with low IPTG and rising aTC concentrations, and a maximal expression rate, exceeding the lower plateau by 20%, when both inducers were present Thus, the synthetic promoter's response to inducers demonstrates characteristics of both OR and AND logic gates, termed a fuzzy AND logic gate.
To understand the fuzzy AND logical behavior of the synthetic promoter, we developed a steady-state mathematical model that incorporates the molecular interactions and their kinetic and thermodynamic properties This model uniquely treats each DNA operator as a distinct chemical species with specific interactions, utilizes the chemical partition function to assess the probability of various regulatory states—including those where both the Holoenzyme and repressor are bound—and distinguishes between inducer-dependent and inducer-independent components This separation enables precise fitting of both the model's "high/low" levels and its response to inducers based on experimental data.
The model predicts how rearranging operator sites in the synthetic promoter affects its activity in response to chemical inducers It does not assume the effectiveness of repressors in preventing holoenzyme assembly, as this would complicate fitting experimental data without relying on arbitrary rate laws Additionally, the model enables the calculation of unknown thermodynamic parameters using only the synthetic promoter's response to chemical inducers, while excluding its undefined minimum and maximum activities Currently, there is no quantitative model available to determine the minimum and maximum expression rates.
CHAPTER 4 CONSTRUCTION, CHARACTERIZATION, AND MATHEMATICAL ANALYSISOFA
The Synthetic Promoter 235 is a constitutive promoter characterized by its sequence, emphasizing the importance of not relying solely on fitting data to unknown or estimated parameters Instead, we focus on the expression rate changes of the promoter in response to varying inducer concentrations The parameters related to this response, including the affinities of repressors for their operators, have been experimentally measured This allows us to substitute known quantities, reduce the degrees of freedom, and calculate the remaining unknown parameters effectively.
Only changes in the steric interaction between the Lac repressor and RNA polymerase at the upstream operator can replicate the experimental data of the synthetic promoter This allows for the calculation of the thermodynamic Gibbs free energy of this interaction with high accuracy, despite being an indirect measurement Directly measuring this positive Gibbs free energy is impractical, as it would involve determining the exact work needed to bring the Lac repressor and RNA polymerase together at chemical equilibrium Nevertheless, we have demonstrated that an accurate indirect measurement is achievable.
Utilizing newly calculated thermodynamic parameters, we apply a mathematical model to predict the inducer response of enhanced synthetic promoters By interchanging the upstream lacO1 and downstream tetO2 operators, we can develop a synthetic promoter that exhibits a more AND-like logical response This approach allows us to initially fit experimental data to determine missing parameters, which can then be leveraged to design novel synthetic promoters with varied functionalities.
Bifurcation analysis of deterministic differential equations reveals how dynamic behaviors change with parameter alterations As we explore smaller physical, chemical, and biological systems, the influence of thermal and mechanical random motion becomes increasingly significant These systems' stochastic dynamics are typically represented as a jump Markov process, governed by a time-dependent probability distribution described by a kinetic Master equation Stochastic bifurcation analysis examines how parameter changes qualitatively affect the steady-state solutions of the kinetic Master equation and the stability of system trajectories in both forward and reverse time.
In this chapter, we employ forward and reverse time stochastic simulation alongside an iterative forward-reverse sampling method to generate the first bifurcation diagram of a stochastic jump Markov process characterized by non-linear transition rates and a potentially infinite number of discrete states By utilizing the bistable chemical Schlégl model, we analyze both stationary and non-stationary steady-state solutions of the forward and reverse kinetic Master equations in relation to a bifurcation parameter These bifurcation diagrams serve as valuable tools for scientists and engineers, aiding in the study and design of systems that experience continuous perturbations from thermal or mechanical noise.
5.2 Conceptual Background on Random Dynamical Systems
A random dynamical system can be illustrated through an example involving a first-order ordinary differential equation, dx/dt = f(x,t), with an initial condition x₀ By numerically integrating this equation over a specified interval [t₀, t₁], we generate a trajectory from x₀ to x₁ If we select a different initial condition and perform the integration again within the same time frame, we produce another trajectory By systematically choosing a fine grid of initial conditions within the domain and repeating the numerical integration for each, we can observe the diverse outcomes that characterize a random dynamical system.
*This assumes existence and uniqueness of the solution