How mathematical modeling could contribute to the quantification of metastatic tumor burden under therapy: insights in immunotherapeutic treatment of non-small cell lung cancer.

BACKGROUND: Cancer is one of the leading death causes globally with about 8.2 million deaths per year and an increase in numbers in recent years. About 90% of cancer deaths do not occur due to primary tumors but due to metastases, of which most are not clinically identifiable because of their relatively small size at primary diagnosis and limited technical possibilities. However, therapeutic decisions are formed depending on the existence of metastases and their properties. Therefore non-identified metastases might have huge influence in the treatment outcome. The quantification of clinically visible and invisible metastases is important for the choice of an optimal treatment of the individual patient as it could clarify the burden of non-identifiable tumors as well as the future behavior of the cancerous disease. RESULTS: The mathematical model presented in this study gives insights in how this could be achieved, taking into account different treatment possibilities and therefore being able to compare therapy schedules for individual patients with different clinical parameters. The framework was tested on three patients with non-small cell lung cancer, one of the deadliest types of cancer worldwide, and clinical history including platinum-based chemotherapy and PD-L1-targeted immunotherapy. Results yield promising insights into the framework to establish methods to quantify effects of different therapy methods and prognostic features for individual patients already at stage of primary diagnosis.

Simulation-Based Exploration of Quorum Sensing Triggered Resistance of Biofilms to Antibiotics.

We present a mathematical model of biofilm response to antibiotics, controlled by a quorum sensing system that confers increased resistance. The model is a highly nonlinear system of partial differential equations that we investigate in computer simulations. Our results suggest that an adaptive, quorum sensing-controlled, mechanism to switch between modes of fast growth with little protection and protective modes of slow growth may confer benefits to biofilm populations. It enhances the formation of micro-niches in the inner regions of the biofilm in which bacteria are not easily reached by antibiotics. Whereas quorum sensing inhibitors can delay the onset of increased resistance, their advantage is lost after up-regulation. This emphasizes the importance of timing for treatment of biofilms with antibiotics.

Analysis of N-acylhomoserine lactone dynamics in continuous cultures of Pseudomonas putida IsoF by use of ELISA and UHPLC/qTOF-MS-derived measurements and mathematical models.

In this interdisciplinary approach, the dynamics of production and degradation of the quorum sensing signal 3-oxo-decanoylhomoserine lactone were studied for continuous cultures of Pseudomonas putida IsoF. The signal concentrations were quantified over time by use of monoclonal antibodies and ELISA. The results were verified by use of ultra-high-performance liquid chromatography. By use of a mathematical model we derived quantitative values for non-induced and induced signal production rate per cell. It is worthy of note that we found rather constant values for different rates of dilution in the chemostat, and the values seemed close to those reported for batch cultures. Thus, the quorum-sensing system in P. putida IsoF is remarkably stable under different environmental conditions. In all chemostat experiments, the signal concentration decreased strongly after a peak, because emerging lactonase activity led to a lower concentration under steady-state conditions. This lactonase activity probably is quorum sensing-regulated. The potential ecological implication of such unique regulation is discussed.

Parameters, architecture and emergent properties of the Pseudomonas aeruginosa LasI/LasR quorum-sensing circuit.

Quorum sensing is a widespread process in bacteria that controls collective behaviours in response to cell density. Populations of cells coordinate gene expression through the perception of self-produced chemical signals. Although this process is well-characterized genetically and biochemically, quantitative information about network properties, including induction dynamics and steady-state behaviour, is scarce. Here we integrate experiments with mathematical modelling to quantitatively analyse the LasI/LasR quorum sensing pathway in the opportunistic pathogen Pseudomonas aeruginosa. We determine key kinetic parameters of the pathway and, using the parametrized model, show that quorum sensing behaves as a bistable hysteretic switch, with stable on and off states. We investigate the significance of feedback architecture and find that positive feedback on signal production is critical for induction dynamics and bistability, whereas positive feedback on receptor expression and negative feedback on signal production play a minor role. Taken together, our data-based modelling approach reveals fundamental and emergent properties of a bacterial quorum sensing circuit, and provides evidence that native quorum sensing can indeed function as the gene expression switch it is commonly perceived to be.

Data suggested hospitalization as critical indicator of the severity of the COVID-19 pandemic, even at its early stages.

COVID-19 has been spreading widely since January 2020, prompting the implementation of non-pharmaceutical interventions and vaccinations to prevent overwhelming the healthcare system. Our study models four waves of the epidemic in Munich over two years using a deterministic, biology-based mathematical model of SEIR type that incorporates both non-pharmaceutical interventions and vaccinations. We analyzed incidence and hospitalization data from Munich hospitals and used a two-step approach to fit the model parameters: first, we modeled incidence without hospitalization, and then we extended the model to include hospitalization compartments using the previous estimates as a starting point. For the first two waves, changes in key parameters, such as contact reduction and increasing vaccinations, were enough to represent the data. For wave three, the introduction of vaccination compartments was essential. In wave four, reducing contacts and increasing vaccinations were critical parameters for controlling infections. The importance of hospitalization data was highlighted, as it should have been included as a crucial parameter from the outset, along with incidence, to avoid miscommunication with the public. The emergence of milder variants like Omicron and a significant proportion of vaccinated people has made this fact even more evident.

A phase-field model for non-small cell lung cancer under the effects of immunotherapy.

Formulating mathematical models that estimate tumor growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our model consists of a Cahn-Hilliard-based phase-field model describing the evolution of proliferative and necrotic tumor cells. These are coupled to a simplified nutrient model that drives the growth of the proliferative cells and their decay into necrotic cells. The applied immunotherapy decreases the proliferative cell concentration. Here, we model the immunotherapeutic agent concentration in the entire lung over time by an ordinary differential equation (ODE). Finally, reaction terms provide a coupling between all these equations. By assuming spherical, symmetric tumor growth and constant nutrient inflow, we simplify this full 3D cancer simulation model to a reduced 1D model. We can then resort to patient data gathered from computed tomography (CT) scans over several years to calibrate our model. Our model covers the case in which the immunotherapy is successful and limits the tumor size, as well as the case predicting a sudden relapse, leading to exponential tumor growth. Finally, we move from the reduced model back to the full 3D cancer simulation in the lung tissue. Thereby, we demonstrate the predictive benefits that a more detailed patient-specific simulation including spatial information as a possible generalization within our framework could yield in the future.

Dynamics of AHL mediated quorum sensing under flow and non-flow conditions.

Quorum sensing (QS) describes the capability of microbes to communicate with each other by the aid of small molecules. Here we investigate the dynamics of QS-regulated gene expression induced by acylhomoserine lactones (AHLs) in Pseudomonas putida IsoF containing a green fluorescent protein-based AHL reporter. The fluorescence time course of individual colonies is monitored following the external addition of a defined AHL concentration to cells which had previously reached the QS-inactive state in AHL-free medium. Using a microfluidic setup the experiment is performed both under flow and non-flow conditions. We find that without supplying external AHL gene expression is induced without flow while flow suppresses the induction. Both without and with flow, at a low AHL concentration the fluorescence onset is significantly delayed while fluorescence starts to increase directly upon the addition of AHL at a high concentration. The differences between no flow and flow can be accounted for using a two-compartment model. This indicates AHL accumulation in a volume which is not affected by the flow. The experiments furthermore show significant cell-to-cell and colony-to-colony variability which is discussed in the context of a compartmentalized QS mechanism.

Dynamics of glucose and insulin concentration connected to the ß-cell cycle: model development and analysis.

BACKGROUND: Diabetes mellitus is a group of metabolic diseases with increased blood glucose concentration as the main symptom. This can be caused by a relative or a total lack of insulin which is produced by the ß-cells in the pancreatic islets of Langerhans. Recent experimental results indicate the relevance of the ß-cell cycle for the development of diabetes mellitus. METHODS: This paper introduces a mathematical model that connects the dynamics of glucose and insulin concentration with the ß-cell cycle. The interplay of glucose, insulin, and ß-cell cycle is described with a system of ordinary differential equations. The model and its development will be presented as well as its mathematical analysis. The latter investigates the steady states of the model and their stability. RESULTS: Our model shows the connection of glucose and insulin concentrations to the ß-cell cycle. In this way the important role of glucose as regulator of the cell cycle and the capability of the ß-cell mass to adapt to metabolic demands can be presented. Simulations of the model correspond to the qualitative behavior of the glucose-insulin regulatory system showed in biological experiments. CONCLUSIONS: This work focus on modeling the physiological situation of the glucose-insulin regulatory system with a detailed consideration of the ß-cell cycle. Furthermore, the presented model allows the simulation of pathological scenarios. Modification of different parameters results in simulation of either type 1 or type 2 diabetes.

Spatial heterogeneity of autoinducer regulation systems.

Autoinducer signals enable coordinated behaviour of bacterial populations, a phenomenon originally described as quorum sensing. Autoinducer systems are often controlled by environmental substances as nutrients or secondary metabolites (signals) from neighbouring organisms. In cell aggregates and biofilms gradients of signals and environmental substances emerge. Mathematical modelling is used to analyse the functioning of the system. We find that the autoinducer regulation network generates spatially heterogeneous behaviour, up to a kind of multicellularity-like division of work, especially under nutrient-controlled conditions. A hybrid push/pull concept is proposed to explain the ecological function. The analysis allows to explain hitherto seemingly contradicting experimental findings.

Environmental stress level to model tumor cell growth and survival.

Survival of living tumor cells underlies many influences such as nutrient saturation, oxygen level, drug concentrations or mechanical forces. Data-supported mathematical modeling can be a powerful tool to get a better understanding of cell behavior in different settings. However, under consideration of numerous environmental factors mathematical modeling can get challenging. We present an approach to model the separate influences of each environmental quantity on the cells in a collective manner by introducing the "environmental stress level". It is an immeasurable auxiliary variable, which quantifies to what extent viable cells would get in a stressed state, if exposed to certain conditions. A high stress level can inhibit cell growth, promote cell death and influence cell movement. As a proof of concept, we compare two systems of ordinary differential equations, which model tumor cell dynamics under various nutrient saturations respectively with and without considering an environmental stress level. Particle-based Bayesian inversion methods are used to quantify uncertainties and calibrate unknown model parameters with time resolved measurements of in vitro populations of liver cancer cells. The calibration results of both models are compared and the quality of fit is quantified. While predictions of both models show good agreement with the data, there is indication that the model considering the stress level yields a better fitting. The proposed modeling approach offers a flexible and extendable framework for considering systems with additional environmental factors affecting the cell dynamics.

A mathematical model of quorum sensing regulated EPS production in biofilm communities.

BACKGROUND: Biofilms are microbial communities encased in a layer of extracellular polymeric substances (EPS). The EPS matrix provides several functional purposes for the biofilm, such as protecting bacteria from environmental stresses, and providing mechanical stability. Quorum sensing is a cell-cell communication mechanism used by several bacterial taxa to coordinate gene expression and behaviour in groups, based on population densities. MODEL: We mathematically model quorum sensing and EPS production in a growing biofilm under various environmental conditions, to study how a developing biofilm impacts quorum sensing, and conversely, how a biofilm is affected by quorum sensing-regulated EPS production. We investigate circumstances when using quorum-sensing regulated EPS production is a beneficial strategy for biofilm cells. RESULTS: We find that biofilms that use quorum sensing to induce increased EPS production do not obtain the high cell populations of low-EPS producers, but can rapidly increase their volume to parallel high-EPS producers. Quorum sensing-induced EPS production allows a biofilm to switch behaviours, from a colonization mode (with an optimized growth rate), to a protection mode. CONCLUSIONS: A biofilm will benefit from using quorum sensing-induced EPS production if bacteria cells have the objective of acquiring a thick, protective layer of EPS, or if they wish to clog their environment with biomass as a means of securing nutrient supply and outcompeting other colonies in the channel, of their own or a different species.

How to Cope with Big Data in Functional Analysis of the Esophagus.

INTRODUCTION: Esophageal motility disorders have a severe impact on patients' quality of life. While high-resolution manometry (HRM) is the gold standard in the diagnosis of esophageal motility disorders, intermittently occurring muscular deficiencies often remain undiscovered if they do not lead to an intense level of discomfort or cause suffering in patients. Ambulatory long-term HRM allows us to study the circadian (dys)function of the esophagus in a unique way. With the prolonged examination period of 24 h, however, there is an immense increase in data which requires personnel and time for evaluation not available in clinical routine. Artificial intelligence (AI) might contribute here by performing an autonomous analysis. METHODS: On the basis of 40 previously performed and manually tagged long-term HRM in patients with suspected temporary esophageal motility disorders, we implemented a supervised machine learning algorithm for automated swallow detection and classification. RESULTS: For a set of 24 h of long-term HRM by means of this algorithm, the evaluation time could be reduced from 3 days to a core evaluation time of 11 min for automated swallow detection and clustering plus an additional 10-20 min of evaluation time, depending on the complexity and diversity of motility disorders in the examined patient. In 12.5% of patients with suggested esophageal motility disorders, AI-enabled long-term HRM was able to reveal new and relevant findings for subsequent therapy. CONCLUSION: This new approach paves the way to the clinical use of long-term HRM in patients with temporary esophageal motility disorders and might serve as an ideal and clinically relevant application of AI.

Interplay of two quorum sensing regulation systems of Vibrio fischeri.

Many bacteria developed a possibility to recognise aspects of their environment or to communicate with each other by chemical signals. An important strategy is the so-called quorum sensing (QS), a regulatory mechanism for the gene expression, where the bacteria measure their own cell density by means of this signalling pathway. One of the best-studied species using QS is the marine luminescent bacterium Vibrio fischeri which is considered here as a model organism. The two main regulatory pathways (lux and ain) are combined to a regulation system, the dynamics is modelled by an ODE system. This system is analysed thoroughly, considering stationary states, dynamical behaviour and the possible biological meaning of it. The influence of different parameter values on the behaviour is examined, the same basic system is able to reflect the peculiarities of different bacteria strains (respectively, their mutants).

Sensitivity of the quorum sensing system is achieved by low pass filtering.

Autoinducer sensing, also known as quorum sensing, is the communication of bacteria by autoinducer (small signaling molecules). Cells respond on extremely low concentrations of autoinducer: only one or two molecules per cell are sufficient. At this signal level a high degree of noise is inherent. We ask for the mechanism that is able to overcome the stochasticity of the signal. By means of a model and parameter fitting we show that the sensing module acts as a low pass filter, representing the biochemical equivalent of a moving average. It is shown that the system works most sensitive in the range of 0-50 nM autoinducer. Moreover, the time scale of the reaction depends on the signal strength in a crucial manner. Nonlinear feedback is able to further enhance the sensitivity. The biological implications of the low pass filter property are discussed.

Transcription, intercellular variability and correlated random walk.

We develop a simple model for the random distribution of a gene product. It is assumed that the only source of variance is due to switching transcription on and off by a random process. Under the condition that the transition rates between on and off are constant we find that the amount of mRNA follows a scaled Beta distribution. Additionally, a simple positive feedback loop is considered. The simplicity of the model allows for an explicit solution also in this setting. These findings in turn allow, e.g., for easy parameter scans. We find that bistable behavior translates into bimodal distributions. These theoretical findings are in line with experimental results.

Pyocyanin Restricts Social Cheating in Pseudomonas aeruginosa.

Quorum sensing (QS) in Pseudomonas aeruginosa coordinates the expression of virulence factors, such as exoproteases and siderophores, that are public goods utilized by the whole population of bacteria, regardless of whether they invested or not in their production. These public goods can be used by QS defective mutants for growth, and since these mutants do not contribute to public goods production, they are considered social cheaters. Pyocyanin is a phenazine that is a toxic, QS-controlled metabolite produced by P. aeruginosa. It is a redox-active compound and promotes the generation of reactive oxygen species; it also possesses antibacterial properties and increases fitness in competition with other bacterial species. Since QS-deficient individuals are less able to tolerate oxidative stress, we hypothesized that the pyocyanin produced by the wild-type population could promote selection of functional QS systems in this bacterium. Here, we demonstrate, using competition experiments and mathematical models, that, indeed, pyocyanin increases the fitness of the cooperative QS-proficient individuals and restricts the appearance of social cheaters. In addition, we also show that pyocyanin is able to select QS in other bacteria such as Acinetobacter baumannii.

The hydrolysis of unsubstituted N-acylhomoserine lactones to their homoserine metabolites. Analytical approaches using ultra performance liquid chromatography.

Derivatives of N-acylhomoserine lactones (HSLs) with different alkanoyl side chains occur as quorum or diffusion sensing molecules in gram-negative bacteria and their quantitative chemical analysis became important as a possible way to follow regulation processes of their pathogenicity towards plants and animals. The lactone-ring of HSLs is chemically and biologically not stable: the corresponding serines can be formed in alkaline conditions and these may presumably behave inactive for the biological system. A fast and MS compatible liquid chromatographic method applying high pressure (ultra performance liquid chromatography) with diode array detection was optimized for the rapid quantitative determination of HSLs and their corresponding hydrolysis products. The technique was used to follow and model the hydrolysis reactions of HSLs as function of pH under controlled conditions. Moreover, the method could be triggered to allow a confirmation in the assignment of the potential HSLs in real samples by analysis of the real samples before and after hydrolysis. Quantitative performance characteristics and the character of the hydrolysis reaction were studied as well. The optimized method was successfully applied to a bacterial culture supernatant real sample containing HSLs.

Cell differentiation defines acute and chronic infection cell types in Staphylococcus aureus.

A central question to biology is how pathogenic bacteria initiate acute or chronic infections. Here we describe a genetic program for cell-fate decision in the opportunistic human pathogen Staphylococcus aureus, which generates the phenotypic bifurcation of the cells into two genetically identical but different cell types during the course of an infection. Whereas one cell type promotes the formation of biofilms that contribute to chronic infections, the second type is planktonic and produces the toxins that contribute to acute bacteremia. We identified a bimodal switch in the agr quorum sensing system that antagonistically regulates the differentiation of these two physiologically distinct cell types. We found that extracellular signals affect the behavior of the agr bimodal switch and modify the size of the specialized subpopulations in specific colonization niches. For instance, magnesium-enriched colonization niches causes magnesium binding to S. aureusteichoic acids and increases bacterial cell wall rigidity. This signal triggers a genetic program that ultimately downregulates the agr bimodal switch. Colonization niches with different magnesium concentrations influence the bimodal system activity, which defines a distinct ratio between these subpopulations; this in turn leads to distinct infection outcomes in vitro and in an in vivo murine infection model. Cell differentiation generates physiological heterogeneity in clonal bacterial infections and helps to determine the distinct infection types.

An approximative approach for single cell spatial modeling of quorum sensing.

Quorum sensing, a special kind of cell-cell communication, has originally been described for well-mixed homogeneous bacterial cultures. However, recent perception supports its ecological relevance for spatially heterogeneous distributed cells, like colonies and biofilms. New experimental techniques allow for single cell analysis under these conditions, which is crucial to understanding the effect of chemical gradients and intercell variations. Based on a reaction-diffusion system, we develop a method that drastically reduces the computational complexity of the model. In comparison to similar former approaches, handling and scaling is much easier. Via a suitable scaling, this approach leads to approximative algebraic equations for the stationary case. This approach can be easily used for numerical situations.

A Mathematical Model of Quorum Sensing Induced Biofilm Detachment.

BACKGROUND: Cell dispersal (or detachment) is part of the developmental cycle of microbial biofilms. It can be externally or internally induced, and manifests itself in discrete sloughing events, whereby many cells disperse in an instance, or in continuous slower dispersal of single cells. One suggested trigger of cell dispersal is quorum sensing, a cell-cell communication mechanism used to coordinate gene expression and behavior in groups based on population densities. METHOD: To better understand the interplay of colony growth and cell dispersal, we develop a dynamic, spatially extended mathematical model that includes biofilm growth, production of quorum sensing molecules, cell dispersal triggered by quorum sensing molecules, and re-attachment of cells. This is a highly nonlinear system of diffusion-reaction equations that we study in computer simulations. RESULTS: Our results show that quorum sensing induced cell dispersal can be an efficient mechanism for bacteria to control the size of a biofilm colony, and at the same time enhance its downstream colonization potential. In fact we find that over the lifetime of a biofilm colony the majority of cells produced are lost into the aqueous phase, supporting the notion of biofilms as cell nurseries. We find that a single quorum sensing based mechanism can explain both, discrete dispersal events and continuous shedding of cells from a colony. Moreover, quorum sensing induced cell dispersal affects the structure and architecture of the biofilm, for example it might lead to the formation of hollow inner regions in a biofilm colony.