Exact methods are available for the simulation of isothermal, well-mixed stochastic chemical kinetics. As increasingly complex physical systems are modeled, however, these methods become difficult to solve because the computational burden scales with the number of reaction events. This paper addresses one aspect of this problem: the case in which reacting species fluctuate by different orders of magnitude. By partitioning the system into subsets of ''fast'' and ''slow'' reactions, it is possible to bound the computational load by approximating ''fast'' reactions either deterministically or as Langevin equations. This paper provides a theoretical background for such approximations and outlines strategies for computing these approximations. Two motivating examples drawn from the fields of particle technology and biotechnology illustrate the accuracy and computational efficiency of these approximations.

State estimators for physical processes often must address different challenges, including nonlinear dynamics, states subject to hard constraints (e.g., nonnegative concentrations), and local optima. In this article, we compare the performance of two such estimators: the extended Kalman filter (EKF) and moving-horizon estimation (MHE). We outline conditions that lead to the formation of multiple optima in the estimator for systems tending to a steady state and propose tests that determine when these conditions hold for chemical reaction networks. Several simulation examples demonstrate estimation failure in the EKF, even in the absence of plantmodel mismatch. We then examine the role that constraints play in determining the performance on these examples of MHE employing local optimization and a "smoothing" update for the arrival cost. This implementation of MHE represents a feasible, on-line alternative to the EKF for industrial practitioners. In each example, the two estimators are given exactly the same information, namely, tuning parameters, model, and measurements; yet MHE consistently provides improved state estimation and greater robustness to both poor guesses of the initial state and tuning parameters in comparison to the EKF. The only price of this improvement is the greater computational expense required to solve the MHE optimization.

The quasi-steady-state approximation ͑QSSA͒ is a model reduction technique used to remove highly reactive species from deterministic models of reaction mechanisms. In many reaction networks the highly reactive intermediates ͑QSSA species͒ have populations small enough to require a stochastic representation. In this work we apply singular perturbation analysis to remove the QSSA species from the chemical master equation for two classes of problems. The first class occurs in reaction networks where all the species have small populations and the QSSA species sample zero the majority of the time. The perturbation analysis provides a reduced master equation in which the highly reactive species can sample only zero, and are effectively removed from the model. The reduced master equation can be sampled with the Gillespie algorithm. This first stochastic QSSA reduction is applied to several example reaction mechanisms ͑including Michaelis-Menten kinetics͒ ͓Biochem. Z. 49, 333 ͑1913͔͒. A general framework for applying the first QSSA reduction technique to new reaction mechanisms is derived. The second class of QSSA model reductions is derived for reaction networks where non-QSSA species have large populations and QSSA species numbers are small and stochastic. We derive this second QSSA reduction from a combination of singular perturbation analysis and the ⍀ expansion. In some cases the reduced mechanisms and reaction rates from these two stochastic QSSA models and the classical deterministic QSSA reduction are equivalent; however, this is not usually the case.

Synthetic circuits offer great promise for generating insights into nature's underlying design principles or forward engineering novel biotechnology applications. However, construction of these circuits is not straightforward. Synthetic circuits generally consist of components optimized to function in their natural context, not in the context of the synthetic circuit. Combining mathematical modeling with directed evolution offers one promising means for addressing this problem. Modeling identifies mutational targets and limits the evolutionary search space for directed evolution, which alters circuit performance without the need for detailed biophysical information. This review examines strategies for integrating modeling and directed evolution and discusses the utility and limitations of available methods.

Bacteria employ quorum sensing, a form of cell-cell communication, to sense changes in population density and regulate gene expression accordingly. This work investigated the rewiring of one quorum-sensing module, the lux circuit from the marine bacterium Vibrio fischeri. Steady-state experiments demonstrate that rewiring the network architecture of this module can yield graded, threshold, and bistable gene expression as predicted by a mathematical model. The experiments also show that the native lux operon is most consistent with a threshold, as opposed to a bistable, response. Each of the rewired networks yielded functional population sensors at biologically relevant conditions, suggesting that this operon is particularly robust. These findings (i) permit prediction of the behaviors of quorum-sensing operons in bacterial pathogens and (ii) facilitate forward engineering of synthetic gene circuits.In bacteria, broadcasting of metabolite or small peptide signaling molecules enables sensing of changes in population density (18,37,57). This mechanism, known as quorum sensing, has been implicated in regulating the virulence factors in a number of bacterial pathogens, such as Vibrio cholerae (61), the bacterium responsible for the severe diarrheal disease cholera; Pseudomonas aeruginosa (30), the opportunistic pathogen responsible for death in cystic fibrosis patients and high mortality rates in immunocompromised individuals; and Staphylococcus aureus (32), a major culprit of infections in surgical wounds. Consequently, improved understanding of quorum-sensing regulation should provide more insight into combating these pathogens by using either traditional chemotherapy or emerging technologies such as quorum-sensing inhibitors (40) or regulated degradation of quorum-sensing signals (13,14).The lux module of the marine bacterium Vibrio fischeri, a facultative symbiont of luminescent fish or squid, serves as one model system for understanding quorum sensing. V. fischeri employs the lux module to regulate gene expression as a function of the population density. The key element of this system is a regulatory cassette consisting of the genes encoding LuxI and LuxR. LuxI is an acylhomoserine lactone (acyl-HSL) synthase; LuxR is a transcriptional regulator activated by the acyl-HSL. The acyl-HSL signaling molecule is produced inside the cell but can freely diffuse across the cell membrane into the environment. Therefore, the acyl-HSL concentration is low at low cell density. As the cell density increases, the signal accumulates in the environment and inside the cell. The signal can then bind LuxR to stabilize the transcription factor, which then activates gene expression. This process, in which activation of gene expression occurs only after attaining a critical cell density, is known as autoinduction.In this work, we sought to computationally and experimentally address the fundamental question of how bacteria regulate cell-cell communication. Quorum sensing is often described in terms of a discrete switch; at low cell den...

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of ͑1͒ partitioning on the basis of fast and slow reactions as opposed to fast and slow species and ͑2͒ conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid ͑deterministic or Langevin equations coupled with discrete stochastic simulation͒ approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.

Rationale: We previously reported that ivacaftor was safe and well tolerated in cohorts aged 12 to <24 months with cystic fibrosis and gating mutations in the ARRIVAL study; here, we report results for cohorts aged 4 to <12 months. Objectives: To evaluate the safety, pharmacokinetics, and pharmacodynamics of ivacaftor in infants aged 4 to <12 months with one or more gating mutations. Methods: ARRIVAL is a single-arm phase 3 study. Infants received 25 mg or 50 mg ivacaftor every 12 hours on the basis of age and weight for 4 days in part A and 24 weeks in part B. Measurements and Main Results: Primary endpoints were safety (parts A and B) and pharmacokinetics (part A). Secondary/tertiary endpoints (part B) included pharmacokinetics and changes in sweat chloride levels, growth, and markers of pancreatic function. Twenty-five infants received ivacaftor, 12 in part A and 17 in part B (four infants participated in both parts). Pharmacokinetics was consistent with that in older groups. Most adverse events were mild or moderate. In part B, cough was the most common adverse event ( n = 10 [58.8%]). Five infants (part A, n = 1 [8.3%]; part B, n = 4 [23.5%]) had serious adverse events, all of which were considered to be not or unlikely related to ivacaftor. No deaths or treatment discontinuations occurred. One infant (5.9%) experienced an alanine transaminase elevation >3 to ≤5× the upper limit of normal at Week 24. No other adverse trends in laboratory tests, vital signs, or ECG parameters were reported. Sweat chloride concentrations and measures of pancreatic obstruction improved. Conclusions: This study of ivacaftor in the first year of life supports treating the underlying cause of cystic fibrosis in children aged ≥4 months with one or more gating mutations. Clinical trial registered with clinicaltrials.gov (NCT02725567).

Background: The CFTR modulator tezacaftor/ivacaftor was efficacious and generally safe and well tolerated in Phase 3 studies in participants ≥12 years of age with cystic fibrosis (CF) homozygous for the F508del-CFTR mutation or heterozygous with a residual function-CFTR mutation (F/F or F /RF respectively). We evaluated tezacaftor/ivacaftor's efficacy and safety over 8 weeks in participants 6 through 11 years of age with these mutations. Methods: Participants were randomized 4:1 to tezacaftor/ivacaftor or a blinding group (placebo for F/F , ivacaftor for F /RF). The primary endpoint was within-group change from baseline in the lung clearance index 2 •5 (LCI 2 •5) through Week 8. Secondary endpoints were change from baseline in sweat chloride (SwCl), cystic fibrosis questionnaire-revised (CFQ-R) respiratory domain score, and safety. Results: Sixty-seven participants received at least one study drug dose. Of those, 54 received tezacaftor/ivacaftor (F/F , 42; F /RF, 12), 10 placebo, and 3 ivacaftor; 66 completed the study. The within-group change in LCI 2 •5 was significantly reduced (improved) by −0 •51 (95% CI: −0 •74, −0 •29). SwCl concentration decreased (improved) by −12 •3 mmol/L and CFQ-R respiratory domain score increased (improved, nonsignificantly) by 2 •3 points. There were no serious adverse events (AEs) or AEs leading to tezacaftor/ivacaftor discontinuation or interruption. The most common AEs (≥10%) in participants receiving tezacaftor/ivacaftor were cough, headache, and productive cough. Conclusions: Tezacaftor/ivacaftor improved lung function (assessed using LCI) and CFTR function (measured by SwCl concentration) in participants 6 through 11 years of age with F/F or F /RF genotypes. Tezacaftor/ivacaftor was safe and well tolerated; no new safety concerns were identified.