arXiv subjects

Steffen Waldherr

Publications and source records attributed to Steffen Waldherr.

At least 19 recordsLinked to original sources

arXiv2025

Modelling the effect of antibody depletion on dose-response behavior for common immunostaining protocols

Antibody binding properties for immunostaining applications are often characterized by dose-response curves, which describe the amount of bound antibodies as a function of the antibody concentration applied at the beginning of the experiment. A common model for the dose-response curve is the Langmuir isotherm, which assumes an equilibrium between the binding and unbinding of antibodies. However, for common immunostaining protocols, the equilibrium assumption is violated, and the dose-response behavior is governed by an accumulation of permanently bound antibodies. Assuming a constant antibody concentration, the resulting accumulation model can easily be solved analytically. However, in many experimental setups the overall amount of antibodies is fixed, such that antibody binding reduces the concentration of free antibodies. Solving the corresponding depletion accumulation model is more difficult and seems to be impossible for heterogeneous epitope landscapes. In this paper, we first solve the depletion-free accumulation model analytically for a homogeneous epitope landscape. From the obtained solution, we derive inequalities between the depletion-free accumulation model, the depletion accumulation model, and the Langmuir isotherm. This allows us to characterize the depletion effect for homogeneous epitope landscapes. Next, we generalize the problem to heterogeneous epitope landscapes, where we prove the existence and uniqueness of a solution that behaves as expected from the experimental setting. These natural properties define bounds for the depletion accumulation model. We conclude this paper by applying the bounds to characterize the depletion effect for heterogeneous epitope landscapes.

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arXiv2024

Protein-environment-sensitive computational epitope accessibility analysis from antibody dose-response data

Antibodies are widely used in life-sciences and medical therapy. Yet, broadly applicable methods are missing to determine, in the biological system of choice, antibody specificity and its quantitative contribution to e.g. immunofluorescence stainings. Thereby, antibody-based data often needs to be seen with caution. Here, we present a simple-to-use approach to characterize and quantify antibody binding properties directly in the system of choice. We determine an epitope accessibility distribution in the system of interest based on a computational analysis of antibody-dilution immunofluorescence stainings. This allows the selection of specific antibodies, the choice of a dilution to maximize signal-specificity, and an improvement of signal quantification. It further expands the scope of antibody-based imaging to detect changes of the subcellular nano-environment and allows for antibody multiplexing.

q-bio.QM↗

arXiv2023

Modeling Cell Size Distribution with Heterogeneous Flux Balance Analysis

For over two decades, Flux Balance Analysis (FBA) has been successfully used for predicting growth rates and intracellular reaction rates in microbiological metabolism. An aspect that is often omitted from this analysis, is segregation or heterogeneity between different cells. In this work, we propose an extended FBA method to model cell size distributions in balanced growth conditions. Hereto, a mathematical description of the concept of balanced growth in terms of cell mass distribution is presented. The cell mass distribution, quantified by the Number Density Function (NDF), is affected by cell growth and cell division. An optimization program is formulated in a general manner in which the NDF, average cell culture growth rate and reaction rates per cell mass are treated as optimization variables. As qualitative proof of concept, the methodology is illustrated on a core carbon model of Escherichia coli under aerobic growth conditions. This illustrates feasibility and applications of this method, while indicating some shortcomings intrinsic to the simplified biomass structuring and the time invariant approach.

eess.SY↗

arXiv2023

COVRECON: Combining Genome-scale Metabolic Network Reconstruction and Data-driven Inverse Modeling to Reveal Changes in Metabolic Interaction Networks

One central goal of systems biology is to infer biochemical regulations from large-scale OMICS data. Many aspects of cellular physiology and organism phenotypes could be understood as a result of the metabolic interaction network dynamics. Previously, we have derived a mathematical method addressing this problem using metabolomics data for the inverse calculation of a biochemical Jacobian network. However, these algorithms for this inference are limited by two issues: they rely on structural network information that needs to be assembled manually, and they are numerically unstable due to ill-conditioned regression problems, which makes them inadequate for dealing with large-scale metabolic networks. In this work, we present a novel regression-loss based inverse Jacobian algorithm and related workflow COVRECON. It consists of two parts: a, Sim-Network and b, Inverse differential Jacobian evaluation. Sim-Network automatically generates an organism-specific enzyme and reaction dataset from Bigg and KEGG databases, which is then used to reconstruct the Jacobian's structure for a specific metabolomics dataset. Instead of directly solving a regression problem, the new inverse differential Jacobian part is based on a more robust approach and rates the biochemical interactions according to their relevance from large-scale metabolomics data. This approach is illustrated by in silico stochastic analysis with different-sized metabolic networks from the BioModels database. The advantages of COVRECON are that 1) it automatically reconstructs a data-driven superpathway metabolic interaction model; 2) more general network structures can be considered; 3) the new inverse algorithms improve stability, decrease computation time, and extend to large-scale models

q-bio.MN↗

arXiv2021

Numerical Gaussian process Kalman filtering for spatiotemporal systems

We present a novel Kalman filter for spatiotemporal systems called the numerical Gaussian process Kalman filter (GPKF). Numerical Gaussian processes have recently been introduced as a physics informed machine learning method for simulating time-dependent partial differential equations without the need for spatial discretization. We bring numerical GPs into probabilistic state space form. This model is linear and its states are Gaussian distributed. These properties enable us to embed the numerical GP state space model into the recursive Kalman filter algorithm. We showcase the method using two case studies.

eess.SY↗

arXiv2020

Numerical Gaussian process Kalman filtering

In this manuscript we introduce numerical Gaussian process Kalman filtering (GPKF). Numerical Gaussian processes have recently been developed to simulate spatiotemporal models. The contribution of this paper is to embed numerical Gaussian processes into the recursive Kalman filter equations. This embedding enables us to do Kalman filtering on infinite-dimensional systems using Gaussian processes. This is possible because i) we are obtaining a linear model from numerical Gaussian processes, and ii) the states of this model are by definition Gaussian distributed random variables. Convenient properties of the numerical GPKF are that no spatial discretization of the model is necessary, and manual setting up of the Kalman filter, that is fine-tuning the process and measurement noise levels by hand is not required, as they are learned online from the data stream. We showcase the capability of the numerical GPKF in a simulation study of the advection equation.

stat.ML↗

arXiv2018

Dynamic Density Estimation in Heterogeneous Cell Populations

Multicellular systems play a key role in bioprocess and biomedical engineering. Cell ensembles encountered in these setups show phenotypic variability like size and biochemical composition. As this variability may result in undesired effects in bioreactors, close monitoring of the cell population heterogeneity is important for maximum production output, and accurate control. However, direct measurements are mostly restricted to a few cellular properties. This motivates the application of model-based online estimation techniques for the reconstruction of non-measurable cellular properties. Population balance modeling allows for a natural description of cell-to-cell variability. In this contribution, we present an estimation approach that, in contrast to existing ones, does not rely on a finite-dimensional approximation through grid based discretization of the underlying population balance model. Instead, our so-called characteristics based density estimator employs sample approximations. With two and three-dimensional benchmark examples we demonstrate that our approach is superior to the grid based designs in terms of accuracy and computational demand.

eess.SY↗

arXiv2018

Dynamic modeling of enzyme controlled metabolic networks using a receding time horizon

Microorganisms have developed complex regulatory features controlling their reaction and internal adaptation to changing environments. When modeling these organisms we usually do not have full understanding of the regulation and rely on substituting it with an optimization problem using a biologically reasonable objective function. The resulting constraint-based methods like the Flux Balance Analysis (FBA) and Resource Balance Analysis (RBA) have proven to be powerful tools to predict growth rates, by-products, and pathway usage for fixed environments. In this work, we focus on the dynamic enzyme-cost Flux Balance Analysis (deFBA), which models the environment, biomass products, and their composition dynamically and contains reaction rate constraints based on enzyme capacity. We extend the original deFBA formalism to include storage molecules and biomass-related maintenance costs. Furthermore, we present a novel usage of the receding prediction horizon as used in Model Predictive Control (MPC) in the deFBA framework, which we call the short-term deFBA (sdeFBA). This way we eliminate some mathematical artifacts arising from the formulation as an optimization problem and gain access to new applications in MPC schemes. A major contribution of this paper is also a systematic approach for choosing the prediction horizon and identifying conditions to ensure solutions grow exponentially. We showcase the effects of using the sdeFBA with different horizons through a numerical example.

math.OC↗

arXiv2017

Modeling metabolic networks including gene expression and uncertainties

Constraint based methods, such as the Flux Balance Analysis, are widely used to model cellular growth processes without relying on extensive information on the regulatory features. The regulation is instead substituted by an optimization problem usually aiming at maximal biomass accumulation. A recent extension to these methods called the dynamic enzyme-cost Flux Balance Analysis (deFBA) is a fully dynamic modeling method allowing for the prediction of necessary enzyme levels under changing environmental conditions. However, this method was designed for deterministic settings in which all dynamics, parameters, etc. are exactly known. In this work, we present a theoretical framework extending the deFBA to handle uncertainties and provide a robust solution. We use the ideas from multi-stage nonlinear Model Predictive Control (MPC) and its feature to represent the evolution of uncertainties by an exponentially growing scenario tree. While this representation is able to construct a deterministic optimization problem in the presence of uncertainties, the computational cost also increases exponentially. We counter this by using a receding prediction horizon and reshape the standard deFBA to the short-time deFBA (sdeFBA). This leads us, along with further simplification of the scenario tree, to the robust deFBA (rdeFBA). This framework is capable of handling the uncertainties in the model itself as well as uncertainties experienced by the modeled system. We applied these algorithms to two case-studies: a minimal enzymatic nutrient uptake network, and the abstraction of the core metabolic process in bacteria.

math.OC↗

arXiv2017

Optimality in cellular storage via the Pontryagin Maximum Principle

We study an optimal control problem arising from a resource allocation problem in cellular metabolism. A minimalistic model that describes the production of enzymatic vs. non-enzymatic biomass components from a single nutrient source is introduced. The basic growth modes with this model are linear growth, where only the non-enzymatic component is produced, and exponential growth with only enzymatic components being produced. Using Pontryagin's maximum principle, we derive the optimal growth trajectory depending on the model's parameters. It turns out that depending on the parameters, either a single growth mode is optimal, or otherwise the optimal solution is a concatenation of exponential growth with linear growth. Importantly, on the short time scale, the choice of growth mode depends only on catalytic rate constants and biomass weights of the two component types, whereas on longer time scales, where the nutrient amount becomes limiting, also the yield coefficients play a role.

math.OC↗

arXiv2015

Ensemble Observability of Linear Systems

We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual view points, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.

math.OC↗

arXiv2014

Dynamic optimization of metabolic networks coupled with gene expression

The regulation of metabolic activity by tuning enzyme expression levels is crucial to sustain cellular growth in changing environments. Metabolic networks are often studied at steady state using constraint-based models and optimization techniques. However, metabolic adaptations driven by changes in gene expression cannot be analyzed by steady state models, as these do not account for temporal changes in biomass composition. Here we present a dynamic optimization framework that integrates the metabolic network with the dynamics of biomass production and composition, explicitly taking into account enzyme production costs and enzymatic capacity. In contrast to the established dynamic flux balance analysis, our approach allows predicting dynamic changes in both the metabolic fluxes and the biomass composition during metabolic adaptations. We applied our algorithm in two case studies: a minimal nutrient uptake network, and an abstraction of core metabolic processes in bacteria. In the minimal model, we show that the optimized uptake rates reproduce the empirical Monod growth for bacterial cultures. For the network of core metabolic processes, the dynamic optimization algorithm predicted commonly observed metabolic adaptations, such as a diauxic switch with a preference ranking for different nutrients, re-utilization of waste products after depletion of the original substrate, and metabolic adaptation to an impending nutrient depletion. These examples illustrate how dynamic adaptations of enzyme expression can be predicted solely from an optimization principle.

q-bio.MN↗

arXiv2014

Generalization of the construction method for multistability-equivalent gene regulatory networks to systems with multi-input multi-output loopbreaking

The problem of equivalence in terms of multistability properties between gene regulatory network models of different dimensionality has been recently addressed by Schittler et al. (2013). The authors in that work proposed construction rules for a high-dimensional dynamical system, when given a low-dimensional dynamical system and the high-dimensional network structure. However, the proof therein was restricted to the class of systems for which all internal feedback loops can be broken by a loopbreaking approach yielding a single-input single-output (SISO) system. In this report, we present the generalization of the proof to systems with any number of internal feedback loops, which will be broken by a generalized loopbreaking approach resulting in a multi-input multi-output (MIMO) system. This generalization of the method renders the construction method applicable to a broad class of gene regulatory network models, thus promoting the transfer of results from core motif models to more realistic, high-dimensional models of gene regulation. We demonstrate the potential and value of our method by applying it to an example of a gene regulatory network in mesenchymal stem cell differentiation.

math.DS↗

arXiv2012

Network-level dynamics of diffusively coupled cells

We study molecular dynamics within populations of diffusively coupled cells under the assumption of fast diffusive exchange. As a technical tool, we propose conditions on boundedness and ultimate boundedness for systems with a singular perturbation, which extend the classical asymptotic stability results for singularly perturbed systems. Based on these results, we show that with common models of intracellular dynamics, the cell population is coordinated in the sense that all cells converge close to a common equilibrium point. We then study a more specific example of coupled cells which behave as bistable switches, where the intracellular dynamics are such that cells may be in one of two equilibrium points. Here, we find that the whole population is bistable in the sense that it converges to a population state where either all cells are close to the one equilibrium point, or all cells are close to the other equilibrium point. Finally, we discuss applications of these results for the robustness of cellular decision making in coupled populations.

q-bio.CB↗

arXiv2012

Efficient parametric analysis of the chemical master equation through model order reduction

Background: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. Results: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. Conclusions: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.

q-bio.QM↗

arXiv2011

Structural requirements and discrimination of cell differentiation networks

Mathematical models of stem cell differentiation are commonly based upon the concept of subsequent cell fate decisions, each controlled by a gene regulatory network. These networks exhibit a multistable behavior and cause the system to switch between qualitatively distinct stable steady states. However, the network structure of such a switching module is often uncertain, and there is lack of knowledge about the exact reaction kinetics. In this paper, we therefore perform an elementary study of small networks consisting of three interacting transcriptional regulators responsible for cell differentiation: We investigate which network structures can reproduce a certain multistable behavior, and how robustly this behavior is realized by each network. In order to approach these questions, we use a modeling framework which only uses qualitative information about the network, yet allows model discrimination as well as to evaluate the robustness of the desired multistability properties. We reveal structural network properties which are necessary and sufficient to realize distinct steady state patterns required for cell differentiation. Our results also show that structural and robustness properties of the networks are related to each other.

q-bio.QM↗

arXiv2010

Searching bifurcations in high-dimensional parameter space via a feedback loop breaking approach

Bifurcations leading to complex dynamical behaviour of non-linear systems are often encountered when the characteristics of feedback circuits in the system are varied. In systems with many unknown or varying parameters, it is an interesting, but difficult problem to find parameter values for which specific bifurcations occur. In this paper, we develop a loop breaking approach to evaluate the influence of parameter values on feedback circuit characteristics. This approach allows a theoretical classification of feedback circuit characteristics related to possible bifurcations in the system. Based on the theoretical results, a numerical algorithm for bifurcation search in a possibly high-dimensional parameter space is developed. The application of the proposed algorithm is illustrated by searching for a Hopf bifurcation in a model of the mitogen activated protein kinase (MAPK) cascade, which is a classical example for biochemical signal transduction.

q-bio.MN↗

arXiv2010

Bridging Time Scales in Cellular Decision Making with a Stochastic Bistable Switch

Cellular transformations which involve a significant phenotypical change of the cell's state use bistable biochemical switches as underlying decision systems. In this work, we aim at linking cellular decisions taking place on a time scale of years to decades with the biochemical dynamics in signal transduction and gene regulation, occuring on a time scale of minutes to hours. We show that a stochastic bistable switch forms a viable biochemical mechanism to implement decision processes on long time scales. As a case study, the mechanism is applied to model the initiation of follicle growth in mammalian ovaries, where the physiological time scale of follicle pool depletion is on the order of the organism's lifespan. We construct a simple mathematical model for this process based on experimental evidence for the involved genetic mechanisms. Despite the underlying stochasticity, the proposed mechanism turns out to yield reliable behavior in large populations of cells subject to the considered decision process. Our model explains how the physiological time constant may emerge from the intrinsic stochasticity of the underlying gene regulatory network. Apart from ovarian follicles, the proposed mechanism may also be of relevance for other physiological systems where cells take binary decisions over a long time scale.

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