Machine learning demonstrates that somatic mutations imprint invariant morphologic features in myelodysplastic syndromes.

Morphologic interpretation is the standard in diagnosing myelodysplastic syndrome (MDS), but it has limitations, such as varying reliability in pathologic evaluation and lack of integration with genetic data. Somatic events shape morphologic features, but the complexity of morphologic and genetic changes makes clear associations challenging. This article interrogates novel clinical subtypes of MDS using a machine-learning technique devised to identify patterns of cooccurrence among morphologic features and genomic events. We sequenced 1079 MDS patients and analyzed bone marrow morphologic alterations and other clinical features. A total of 1929 somatic mutations were identified. Five distinct morphologic profiles with unique clinical characteristics were defined. Seventy-seven percent of higher-risk patients clustered in profile 1. All lower-risk (LR) patients clustered into the remaining 4 profiles: profile 2 was characterized by pancytopenia, profile 3 by monocytosis, profile 4 by elevated megakaryocytes, and profile 5 by erythroid dysplasia. These profiles could also separate patients with different prognoses. LR MDS patients were classified into 8 genetic signatures (eg, signature A had TET2 mutations, signature B had both TET2 and SRSF2 mutations, and signature G had SF3B1 mutations), demonstrating association with specific morphologic profiles. Six morphologic profiles/genetic signature associations were confirmed in a separate analysis of an independent cohort. Our study demonstrates that nonrandom or even pathognomonic relationships between morphology and genotype to define clinical features can be identified. This is the first comprehensive implementation of machine-learning algorithms to elucidate potential intrinsic interdependencies among genetic lesions, morphologies, and clinical prognostic in attributes of MDS.

A numerical framework for computing steady states of structured population models and their stability.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

A numerical framework for computing steady states of structured population models and their stability.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations.

We investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We formulate the inverse problem as a least squares problem for the probability measure estimation. Using the Prohorov metric framework, we prove existence and consistency of the least squares estimates and outline a discretization scheme for approximating a conditional probability measure. For this scheme, we prove general method stability. The work is motivated by Partial Differential Equation (PDE) models of flocculation for which the shape of the post-fragmentation conditional probability measure greatly impacts the solution dynamics. To illustrate our methodology, we apply the theory to a particular PDE model that arises in the study of population dynamics for flocculating bacterial aggregates in suspension, and provide numerical evidence for the utility of the approach.

Laplacian Dynamics with Synthesis and Degradation.

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first-order degradation and zeroth-order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer an open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of such a matrix. This also completes our previous framework and makes it purely graph theoretical. Lastly, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels in pancreatic ß-cells.