Conservation Laws and Symmetries in Competitive Systems

We investigate a conservation law of a system of symmetric 2n-dimensional nonlinear differential equations. We use Lagrangian approach and Noether's theorem to analyze Lotka-Volterra type of competitive system. We observe that the coefficients of the 2n-dimensional nonlinear differential equations are strictly restricted when the system has a conserved quantity, and the relation between a conserved system and Lyapunov function is shown in terms of Noether's theorem. We find that a system of the 2n-dimensional first-order nonlinear differential equations in a symmetric form should appear in a binary-coupled form (BCF), and a BCF has a conserved quantity if parameters satisfy certain conditions. The conservation law manifests characteristic properties of a system of nonlinear differential equations and can be employed to check the accuracy of numerical solutions in the BCF

Sector dominance ratio analysis of financial markets

In this paper we present a new measure to investigate the functional structure of financial markets, the Sector Dominance Ratio (SDR). We study the information embedded in raw and partial correlations using random matrix theory (RMT) and examine the evolution of economic sectoral makeup on a yearly and monthly basis for four stock markets, those of the US, UK, Germany and Japan, during the period from January 2000 to December 2010. We investigate the information contained in raw and partial correlations using the sector dominance ratio and its variation over time. The evolution of economic sectoral activities can be discerned through the largest eigenvectors of both raw correlation and partial correlation matrices. We find a characteristic change of the largest eigenvalue from raw and partial correlations and the SDR that coincides with sharp breaks in asset valuations. Finally, we propose the SDR as an indicator for changes in VIX indexes

Transcriptome free energy can serve as a dynamic patient-specific biomarker in acute myeloid leukemia

Acute myeloid leukemia (AML) is prevalent in both adult and pediatric patients. Despite advances in patient categorization, the heterogeneity of AML remains a challenge. Recent studies have explored the use of gene expression data to enhance AML diagnosis and prognosis, however, alternative approaches rooted in physics and chemistry may provide another level of insight into AML transformation. Utilizing publicly available databases, we analyze 884 human and mouse blood and bone marrow samples. We employ a personalized medicine strategy, combining state-transition theory and surprisal analysis, to assess the RNA transcriptome of individual patients. The transcriptome is transformed into physical parameters that represent each sample's steady state and the free energy change (FEC) from that steady state, which is the state with the lowest free energy.We found the transcriptome steady state was invariant across normal and AML samples. FEC, representing active molecular processes, varied significantly between samples and was used to create patient-specific barcodes to characterize the biology of the disease. We discovered that AML samples that were in a transition state had the highest FEC. This disease state may be characterized as the most unstable and hence the most therapeutically targetable since a change in free energy is a thermodynamic requirement for disease progression. We also found that distinct sets of ongoing processes may be at the root of otherwise similar clinical phenotypes, implying that our integrated analysis of transcriptome profiles may facilitate a personalized medicine approach to cure AML and restore a steady state in each patient

Dynamic patterns of microRNA expression during acute myeloid leukemia state-transition.

MicroRNAs (miRNAs) have been shown to hold prognostic value in acute myeloid leukemia (AML); however, the temporal dynamics of miRNA expression in AML are poorly understood. Using serial samples from a mouse model of AML to generate time-series miRNA sequencing data, we are the first to show that the miRNA transcriptome undergoes state-transition during AML initiation and progression. We modeled AML state-transition as a particle undergoing Brownian motion in a quasi-potential and validated the AML state-space and state-transition model to accurately predict time to AML in an independent cohort of mice. The critical points of the model provided a framework to align samples from mice that developed AML at different rates. Our mathematical approach allowed discovery of dynamic processes involved during AML development and, if translated to humans, has the potential to predict an individuals disease trajectory

Proteomics and mathematical modeling of longitudinal CSF differentiates fast versus slow ALS progression

Abstract Objective Amyotrophic lateral sclerosis (ALS) is a heterogeneous disease with a complex etiology that lacks biomarkers predicting disease progression. The objective of this study was to use longitudinal cerebrospinal fluid (CSF) samples to identify biomarkers that distinguish fast progression (FP) from slow progression (SP) and assess their temporal response. Methods We utilized mass spectrometry (MS)‐based proteomics to identify candidate biomarkers using longitudinal CSF from a discovery cohort of SP and FP ALS patients. Immunoassays were used to quantify and validate levels of the top biomarkers. A state‐transition mathematical model was created using the longitudinal MS data that also predicted FP versus SP. Results We identified a total of 1148 proteins in the CSF of all ALS patients. Pathway analysis determined enrichment of pathways related to complement and coagulation cascades in FPs and synaptogenesis and glucose metabolism in SPs. Longitudinal analysis revealed a panel of 59 candidate markers that could segregate FP and SP ALS. Based on multivariate analysis, we identified three biomarkers (F12, RBP4, and SERPINA4) as top candidates that segregate ALS based on rate of disease progression. These proteins were validated in the discovery and a separate validation cohort. Our state‐transition model determined that the overall variance of the proteome over time was predictive of the disease progression rate. Interpretation We identified pathways and protein biomarkers that distinguish rate of ALS disease progression. A mathematical model of the CSF proteome determined that the change in entropy of the proteome over time was predictive of FP versus SP

Proteomics and mathematical modeling of longitudinal CSF differentiates fast versus slow ALS progression

ObjectiveAmyotrophic lateral sclerosis (ALS) is a heterogeneous disease with a complex etiology that lacks biomarkers predicting disease progression. The objective of this study was to use longitudinal cerebrospinal fluid (CSF) samples to identify biomarkers that distinguish fast progression (FP) from slow progression (SP) and assess their temporal response.MethodsWe utilized mass spectrometry (MS)-based proteomics to identify candidate biomarkers using longitudinal CSF from a discovery cohort of SP and FP ALS patients. Immunoassays were used to quantify and validate levels of the top biomarkers. A state-transition mathematical model was created using the longitudinal MS data that also predicted FP versus SP.ResultsWe identified a total of 1148 proteins in the CSF of all ALS patients. Pathway analysis determined enrichment of pathways related to complement and coagulation cascades in FPs and synaptogenesis and glucose metabolism in SPs. Longitudinal analysis revealed a panel of 59 candidate markers that could segregate FP and SP ALS. Based on multivariate analysis, we identified three biomarkers (F12, RBP4, and SERPINA4) as top candidates that segregate ALS based on rate of disease progression. These proteins were validated in the discovery and a separate validation cohort. Our state-transition model determined that the overall variance of the proteome over time was predictive of the disease progression rate.InterpretationWe identified pathways and protein biomarkers that distinguish rate of ALS disease progression. A mathematical model of the CSF proteome determined that the change in entropy of the proteome over time was predictive of FP versus SP

State-transition modeling of blood transcriptome predicts disease evolution and treatment response in chronic myeloid leukemia

Chronic myeloid leukemia (CML) is initiated and maintained by BCR::ABL which is clinically targeted using tyrosine kinase inhibitors (TKIs). TKIs can induce long-term remission but are also not curative. Thus, CML is an ideal system to test our hypothesis that transcriptome-based state-transition models accurately predict cancer evolution and treatment response. We collected time-sequential blood samples from tetracycline-off (Tet-Off) BCR::ABL-inducible transgenic mice and wild-type controls. From the transcriptome, we constructed a CML state-space and a three-well leukemogenic potential landscape. The potential's stable critical points defined observable disease states. Early states were characterized by anti-CML genes opposing leukemia; late states were characterized by pro-CML genes. Genes with expression patterns shaped similarly to the potential landscape were identified as drivers of disease transition. Re-introduction of tetracycline to silence the BCR::ABL gene returned diseased mice transcriptomes to a near healthy state, without reaching it, suggesting parts of the transition are irreversible. TKI only reverted the transcriptome to an intermediate disease state, without approaching a state of health; disease relapse occurred soon after treatment. Using only the earliest time-point as initial conditions, our state-transition models accurately predicted both disease progression and treatment response, supporting this as a potentially valuable approach to time clinical intervention, before phenotypic changes become detectable