Order Reduction of the Chemical Master Equation via Balanced Realisation

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than the Finite State Projection Algorithm (FSP) when used for obtaining transient responses. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. In such an example, we obtain an approximation of the output of a model with 301 states by a reduced model with 10 states. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product; and compare its dynamics with a stochastic Michaelis-Menten representation. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Finally, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules.Comment: 12 pages, 6 figure

Positive Feedback in the AKT/mTOR pathway and its implications for growth signal progression in skeletal muscle cells: An analytical study

The IGF-1 mediated AKT/mTOR pathway has been recently proposed as mediator of skeletal muscle growth and a positive feedback between Akt and mTOR was suggested to induce homogenous growth signals along the whole spatial extension of such long cells. Here we develop two biologically justied approximations which we study under the presence of four dierent initial conditions that describe dierent paradigms of IGF-1 receptor{induced Akt/mTOR activation. In rst scenario the activation of the feedback cascade was assumed to be mild or protein turnover considered to be high. In turn, in the second scenario the transcriptional regulation was assumed to maintain dened levels of inactive pro{enzymes. For both scenarios, we were able to obtain closed{form formulas for growth signal progression in time and space and found that a localised initial signal maintains its Gaussian shape, but gets delocalised and exponentially degraded. Importantly, mathematical treatment of the reaction diusion system revealed that diusion ltered out high frequencies of spatially periodic initiator signals suggesting that the muscle cell is robust against uctuations in spatial receptor expression or activation. However, neither scenario was consistent with the presence of stably travelling signal waves. Our study highlights the role of feedback loops in spatiotemporal signal progression and results can be applied to studies in cell proliferation, cell dierentiation and cell death in other spatially extended cells

Mortality from gastrointestinal congenital anomalies at 264 hospitals in 74 low-income, middle-income, and high-income countries: a multicentre, international, prospective cohort study

Summary Background Congenital anomalies are the fifth leading cause of mortality in children younger than 5 years globally. Many gastrointestinal congenital anomalies are fatal without timely access to neonatal surgical care, but few studies have been done on these conditions in low-income and middle-income countries (LMICs). We compared outcomes of the seven most common gastrointestinal congenital anomalies in low-income, middle-income, and high-income countries globally, and identified factors associated with mortality. Methods We did a multicentre, international prospective cohort study of patients younger than 16 years, presenting to hospital for the first time with oesophageal atresia, congenital diaphragmatic hernia, intestinal atresia, gastroschisis, exomphalos, anorectal malformation, and Hirschsprung’s disease. Recruitment was of consecutive patients for a minimum of 1 month between October, 2018, and April, 2019. We collected data on patient demographics, clinical status, interventions, and outcomes using the REDCap platform. Patients were followed up for 30 days after primary intervention, or 30 days after admission if they did not receive an intervention. The primary outcome was all-cause, in-hospital mortality for all conditions combined and each condition individually, stratified by country income status. We did a complete case analysis. Findings We included 3849 patients with 3975 study conditions (560 with oesophageal atresia, 448 with congenital diaphragmatic hernia, 681 with intestinal atresia, 453 with gastroschisis, 325 with exomphalos, 991 with anorectal malformation, and 517 with Hirschsprung’s disease) from 264 hospitals (89 in high-income countries, 166 in middleincome countries, and nine in low-income countries) in 74 countries. Of the 3849 patients, 2231 (58·0%) were male. Median gestational age at birth was 38 weeks (IQR 36–39) and median bodyweight at presentation was 2·8 kg (2·3–3·3). Mortality among all patients was 37 (39·8%) of 93 in low-income countries, 583 (20·4%) of 2860 in middle-income countries, and 50 (5·6%) of 896 in high-income countries (p<0·0001 between all country income groups). Gastroschisis had the greatest difference in mortality between country income strata (nine [90·0%] of ten in lowincome countries, 97 [31·9%] of 304 in middle-income countries, and two [1·4%] of 139 in high-income countries; p≤0·0001 between all country income groups). Factors significantly associated with higher mortality for all patients combined included country income status (low-income vs high-income countries, risk ratio 2·78 [95% CI 1·88–4·11], p<0·0001; middle-income vs high-income countries, 2·11 [1·59–2·79], p<0·0001), sepsis at presentation (1·20 [1·04–1·40], p=0·016), higher American Society of Anesthesiologists (ASA) score at primary intervention (ASA 4–5 vs ASA 1–2, 1·82 [1·40–2·35], p<0·0001; ASA 3 vs ASA 1–2, 1·58, [1·30–1·92], p<0·0001]), surgical safety checklist not used (1·39 [1·02–1·90], p=0·035), and ventilation or parenteral nutrition unavailable when needed (ventilation 1·96, [1·41–2·71], p=0·0001; parenteral nutrition 1·35, [1·05–1·74], p=0·018). Administration of parenteral nutrition (0·61, [0·47–0·79], p=0·0002) and use of a peripherally inserted central catheter (0·65 [0·50–0·86], p=0·0024) or percutaneous central line (0·69 [0·48–1·00], p=0·049) were associated with lower mortality. Interpretation Unacceptable differences in mortality exist for gastrointestinal congenital anomalies between lowincome, middle-income, and high-income countries. Improving access to quality neonatal surgical care in LMICs will be vital to achieve Sustainable Development Goal 3.2 of ending preventable deaths in neonates and children younger than 5 years by 2030

Contributions to the Analysis of Biochemical Reaction-Diffusion Networks Stability, Analysis, and Numerical Solutions

In this thesis we address dynamic systems problems that arise from the study of biochemical networks. Here we prefer a rigorous treatment of the differential equations that govern their spatio-temporal dynamics, at the cost of studying simplified scenarios of the biological systems under study. Although these simplified scenarios do not model all aspects of the complex interplay in the biological system, they are derived to study the relationship between specific causes and effects. However, by abstracting the systems under study, we obtain the benefit of having models that represent a large variety of processes. For instance, a simple activation mechanism studied here may be used to model the autoactivation of the effector caspase in the apoptosis pathway, the activation of the Akt/mTOR complex implicated in muscular growth, and twospecies population dynamics. In particular, we derive analytical expressions for the equilibrium points of a circular protein activation mechanism with an arbitrary number of intermediate steps and characterise its local stability. Later we analyse the signalling progression due to a protein autoactivation in a long cell. Furthermore, we avail of a projection method for partial differential equations to obtain associated ordinary differential equations that will assist on the reduction of the computational load for the numerical solution of a class of reaction diffusion networks. This projection method will also be used to compute the time-integral of some species concentration in a class of reaction-diffusion networks. Since we chose a theoretical approach, our results provide analytical expressions that link the kinetic parameters and topology of the reaction network with its dynamical behaviour. These formulas can be further studied to analyse the sensitivity of the systems characteristic with respect to variation of parameters as well as explicitly unveiling the main processes that affect the features of interest. We believe that these theoretical approaches provide a deeper insight in selected biochemical pathways such as: the Akt/mTOR activation pathway, mediated by the IGF receptor; the core apoptosis pathway; and Ca2+ homeostasis in non-excitable cells

Equilibria and Stability of a Class of Positive Feedback Loops : Mathematical analysis and its application to caspase-dependent apoptosis

Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops deter- mines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain The- orem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biolog- ical meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability

Positive Feedback in the AKT/mTOR pathway and its implications for growth signal progression in skeletal muscle cells: An analytical study

The IGF-1 mediated AKT/mTOR pathway has been recently proposed as mediator of skeletal muscle growth and a positive feedback between Akt and mTOR was suggested to induce homogenous growth signals along the whole spatial extension of such long cells. Here we develop two biologically justied approximations which we study under the presence of four dierent initial conditions that describe dierent paradigms of IGF-1 receptor{induced Akt/mTOR activation. In rst scenario the activation of the feedback cascade was assumed to be mild or protein turnover considered to be high. In turn, in the second scenario the transcriptional regulation was assumed to maintain dened levels of inactive pro{enzymes. For both scenarios, we were able to obtain closed{form formulas for growth signal progression in time and space and found that a localised initial signal maintains its Gaussian shape, but gets delocalised and exponentially degraded. Importantly, mathematical treatment of the reaction diusion system revealed that diusion ltered out high frequencies of spatially periodic initiator signals suggesting that the muscle cell is robust against uctuations in spatial receptor expression or activation. However, neither scenario was consistent with the presence of stably travelling signal waves. Our study highlights the role of feedback loops in spatiotemporal signal progression and results can be applied to studies in cell proliferation, cell dierentiation and cell death in other spatially extended cells

Western countries and Oil factor in the 70s of 20th century.

Darba izpētes mērķis ir, pamatojoties uz vēstures avotiem un literatūru, izanalizēt un atklāt, kādu iespaidu atstāja naftas faktors uz starptautisko situāciju un Rietumvalstu ekonomiku, politiku un sabiedrisko noskaņojumu 20. gadsimta 70. gados. Darba hronoloģiskās robežas aptver laika posmu no 1971. gada 14. februāra, kad tika noslēgta Teherānas vienošanās starp OPEC un naftas kompānijām, līdz 1979. gadam, kad notikumi Irānā izsauca otro enerģisko krīzi un naftas cenu kāpumu. Pētījumā uzsvars tiek likts uz 1973. gadā pirmo enerģisko krīzi un tās sekām, kā arī uz Rietumvalstu, īpaši ASV, skatījumu uz OPEC arābu valstu pielietoto „naftas ieroci”. Pētījuma ietvaros konstatēts, ka OPEC arābu valstu rīcību motīvi Rietumvalstīs netika pareizi saprasti, savukārt, OPEC loma 70. gados Rietumvalstīs tika pārvērtēta.In final Master work’s aim is to analyze and discover the impact the oil factor left on international situation and economics, policies and public sentiment in Western countries during the 70s of 20th century. Work chronological boundaries cover the period from February 14, 1971, when Tehran agreements between OPEC and oil companies was signed, till 1979, when events in Iran caused the second energy crisis and oil prices peak. The research emphasizes the first energy crisis in 1973 and its consequences, as well as Western countries, particularly the U.S., views on OPEC Arab countries applied „oil weapon”. Research concludes that the motives of OPEC Arab countries were not properly understood in Western countries, and OPEC’s role in the 70s has beed overestimated in the West

Positive Feedback in the AKT/mTOR pathway and its implications for growth signal progression in skeletal muscle cells: An analytical study

The IGF-1 mediated AKT/mTOR pathway has been recently proposed as mediator of skeletal muscle growth and a positive feedback between Akt and mTOR was suggested to induce homogenous growth signals along the whole spatial extension of such long cells. Here we develop two biologically justied approximations which we study under the presence of four dierent initial conditions that describe dierent paradigms of IGF-1 receptor{induced Akt/mTOR activation. In rst scenario the activation of the feedback cascade was assumed to be mild or protein turnover considered to be high. In turn, in the second scenario the transcriptional regulation was assumed to maintain dened levels of inactive pro{enzymes. For both scenarios, we were able to obtain closed{form formulas for growth signal progression in time and space and found that a localised initial signal maintains its Gaussian shape, but gets delocalised and exponentially degraded. Importantly, mathematical treatment of the reaction diusion system revealed that diusion ltered out high frequencies of spatially periodic initiator signals suggesting that the muscle cell is robust against uctuations in spatial receptor expression or activation. However, neither scenario was consistent with the presence of stably travelling signal waves. Our study highlights the role of feedback loops in spatiotemporal signal progression and results can be applied to studies in cell proliferation, cell dierentiation and cell death in other spatially extended cells

Positive Feedback in the AKT/mTOR pathway and its implications for growth signal progression in skeletal muscle cells: An analytical study

The IGF-1 mediated AKT/mTOR pathway has been recently proposed as mediator of skeletal muscle growth and a positive feedback between Akt and mTOR was suggested to induce homogenous growth signals along the whole spatial extension of such long cells. Here we develop two biologically justied approximations which we study under the presence of four dierent initial conditions that describe dierent paradigms of IGF-1 receptor{induced Akt/mTOR activation. In rst scenario the activation of the feedback cascade was assumed to be mild or protein turnover considered to be high. In turn, in the second scenario the transcriptional regulation was assumed to maintain dened levels of inactive pro{enzymes. For both scenarios, we were able to obtain closed{form formulas for growth signal progression in time and space and found that a localised initial signal maintains its Gaussian shape, but gets delocalised and exponentially degraded. Importantly, mathematical treatment of the reaction diusion system revealed that diusion ltered out high frequencies of spatially periodic initiator signals suggesting that the muscle cell is robust against uctuations in spatial receptor expression or activation. However, neither scenario was consistent with the presence of stably travelling signal waves. Our study highlights the role of feedback loops in spatiotemporal signal progression and results can be applied to studies in cell proliferation, cell dierentiation and cell death in other spatially extended cells