Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtolitres. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small sub-cellular compartment. This is achieved by applying a mesoscopic version of the quasi-steady state assumption to the exact Fokker-Planck equation associated with the Poisson Representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing sub-cellular volume, decreasing Michaelis-Menten constants and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.Comment: 13 pages, 4 figures; published in The Journal of Chemical Physic

Modifications of the Levi core

We construct a family of subdistributions of the Levi core $\mathfrak{C}(\mathcal{N})$ called modified Levi cores $\{\mathcal{M}\mathfrak{C}_{\mathcal{A}}\}_{\mathcal{A}}$ indexed over closed distributions $\mathcal{A}$ that contain the Levi null distribution $\mathcal{N}$ and are contained in the complex tangent bundle $T^{1, 0}b\Omega$ of a smooth bounded pseudoconvex domain $\Omega$. We show that Catlin's Property ($P$) holds on $b\Omega$ if and only if Property ($P$) holds on the support of one, and hence all, of the modified Levi cores. In $\mathbb{C}^2$, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in $\mathbb{C}^2$ that satisfies Property ($P$), we show that its modified Levi core is trivial. This contrasts with $\mathfrak{C}(\mathcal{N})$, which can be nontrivial for such domains.Comment: 13 page

How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy

Strong Stein neighborhood bases

Let D be a smooth bounded pseudoconvex domain in C^n. We give several characterizations for the closure of D to have a strong Stein neighborhood basis in the sense that D has a defining function r such that {z\in C^n:r(z)<a} is pseudoconvex for sufficiently small a>0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly to the boundary.Comment: 14 pages, fixed same references, to appear in Complex Var. Elliptic Eq

Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators

The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA) which is a reduced version of the LNA under conditions of timescale separation. In this paper, we present the first rigorous derivation of the ssLNA using the projection operator technique and show that the ssLNA follows uniquely from the standard LNA under the same conditions of timescale separation as those required for the deterministic quasi-steady state approximation. We also show that the large molecule number limit of several common stochastic model reduction techniques under timescale separation conditions constitutes a special case of the ssLNA.Comment: 10 pages, 1 figure, submitted to Physical Review E; see also BMC Systems Biology 6, 39 (2012

High diagnostic stability of confirmed migraine and confirmed tension-type headache according to the ICHD-3 beta in adolescents

Background: Stable headache diagnosis classification is a prerequisite for identification of headache type specific risk factors. Does the stability of a headache diagnosis over time vary between migraine and tension-type headache (TTH)? Are there differences in diagnosis stability between a probable and a definite headache diagnosis? Findings: In a sample of 783 students (ages 12 to 18 years) participating in a headache intervention study in greater Munich, the stability of headache classification according to the International Classification of Headache Disorder - third edition (beta version) (ICHD-3 beta) after a follow-up of 7 months was examined. Differences in stability of probable or definite migraine and probable or definite TTH were assessed. The stability of the headache diagnosis was assessed as predictive value of headache diagnosis with regard to confirmation of the headache type using the same diagnostic instrument 7 months later. Predictive values with 95% confidence intervals (CI) are reported. Of students with initial migraine, a diagnosis of migraine was confirmed in 65.71% of students after 7 months (95%-CI {[}59.40-71.64]). A clear distinction between probable (44.71%, 95%-CI {[}33.91-53.89]) and confirmed diagnosis (76.88% 95%-CI {[}69.56-83.17]) of migraine was observed. For TTH the predictive value was 62.66% (95%-CI {[}57.07-68.01]) overall with a lower stability for probable (46.10%, 95%-CI {[}37.68-54.69]) compared to the confirmed diagnosis (69.71%, 95%-CI {[}23.58-37.67]). Conclusion: While confirmed migraine and confirmed TTH diagnoses seem stable over time, stability of a probable diagnosis for either headache type was lower

Self-reported neck and shoulder pain in adolescents is associated with episodic and chronic migraine

Aim: The aim of this study was to verify the association between self-reported neck/shoulder pain and migraine and to compare findings of chronic and episodic migraine in adolescents. Methods In this cross-sectional study, 601 secondary-school students filled in questionnaires about headache appearance, type and frequency, neck and shoulder pain and lifestyle factors. Results: The adjusted strength of the association between reported neck and shoulder pain and migraine (assessed in multinomial regression models) increased with the frequency of migraine: less than once a week (OR=1.40;95% CI=(0.85-2.30)), weekly (OR=2.14;95% CI=(1.42-3.24)), and at least 15 days/month (OR=7.27;95% CI=(3.42-15.44)). Conclusion: In adolescents the association between self-reported neck and shoulder pain and migraine is most pronounced in migraine with a high attack frequency

Migraine and tension type headache in adolescents at grammar school in Germany - burden of disease and health care utilization

Background Tension-type headache and migraine are among the most prevalent chronic disorders in children/adolescents. Data on health care utilization for headache in this age group, however, are sparse. Methods In 1399 grammar school students (aged 12–19 years) with headache in the last six months in Germany a) the burden of disease for headache (mean intensity, mean frequency in the last three months and PedMIDAS means), b) medical care utilization defined by proportion of students consulting a physician in the last 12 months and/or taking analgetic drugs in the last three months by headache types (migraine and tension-type headache) and by burden of disease were assessed. Results Primary headache substantially impaired daily living activities in adolescents which was mainly related to migraine. Medical care utilization and drug use, however, was low (consulting a physician: 12.0 %, 95 %-CI = [10.3-13.8]; taking analgetic drugs: 29.9 %, 95 %-CI = [27.5-32.4]) - even among students with severe headache (physician consultation: <35 %; taking analgetic drugs: <63 %). Two thirds of students with any headache and 40 % of those with migraine had neither seen a physician nor used analgetic drugs because of their headache in the preceding 12 months. Conclusions Adolescents with headache might too rarely seek professional help for treatment of headache. Health promotion in adolescents should increase awareness for evidence-based treatment options for headache