Mathematical expansion and clinical application of chronic kidney disease stage as vector field.

There are cases in which CKD progression is difficult to evaluate, because the changes in estimated glomerular filtration rate (eGFR) and proteinuria sometimes show opposite directions as CKD progresses. Indices and models that enable the easy and accurate risk prediction of end-stage-kidney disease (ESKD) are indispensable to CKD therapy. In this study, we investigated whether a CKD stage coordinate transformed into a vector field (CKD potential model) accurately predicts ESKD risk. Meta-analysis of large-scale cohort studies of CKD patients in PubMed was conducted to develop the model. The distance from CKD stage G2 A1 to a patient's data on eGFR and proteinuria was defined as r. We developed the CKD potential model on the basis of the data from the meta-analysis of three previous cohort studies: ESKD risk = exp(r). Then, the model was validated using data from a cohort study of CKD patients in Japan followed up for three years (n = 1,564). Moreover, the directional derivative of the model was developed as an index of CKD progression velocity. For ESKD prediction in three years, areas under the receiver operating characteristic curves (AUCs) were adjusted for baseline characteristics. Cox proportional hazards models with spline terms showed the exponential association between r and ESKD risk (p<0.0001). The CKD potential model more accurately predicted ESKD with an adjusted AUC of 0.81 (95% CI 0.76, 0.87) than eGFR (p<0.0001). Moreover, the directional derivative of the model showed a larger adjusted AUC for the prediction of ESKD than the percent eGFR change and eGFR slope (p<0.0001). Then, a chart of the transformed CKD stage was developed for implementation in clinical settings. This study indicated that the transformed CKD stage as a vector field enables the easy and accurate estimation of ESKD risk and CKD progression and suggested that vector analysis is a useful tool for clinical studies of CKD and its related diseases.

New marker for chronic kidney disease progression and mortality in medical-word virtual space.

A new marker reflecting the pathophysiology of chronic kidney disease (CKD) has been desired for its therapy. In this study, we developed a virtual space where data in medical words and those of actual CKD patients were unified by natural language processing and category theory. A virtual space of medical words was constructed from the CKD-related literature (n = 165,271) using Word2Vec, in which 106,612 words composed a network. The network satisfied vector calculations, and retained the meanings of medical words. The data of CKD patients of a cohort study for 3 years (n = 26,433) were transformed into the network as medical-word vectors. We let the relationship between vectors of patient data and the outcome (dialysis or death) be a marker (inner product). Then, the inner product accurately predicted the outcomes: C-statistics of 0.911 (95% CI 0.897, 0.924). Cox proportional hazards models showed that the risk of the outcomes in the high-inner-product group was 21.92 (95% CI 14.77, 32.51) times higher than that in the low-inner-product group. This study showed that CKD patients can be treated as a network of medical words that reflect the pathophysiological condition of CKD and the risks of CKD progression and mortality.

Practical guide to using Kendall's τ in the context of forecasting critical transitions.

Recent studies demonstrate that trends in indicators extracted from measured time series can indicate an approach of an impending transition. Kendall's τ coefficient is often used to study the trend of statistics related to the critical slowing down phenomenon and other methods to forecast critical transitions. Because statistics are estimated from time series, the values of Kendall's τ are affected by parameters such as window size, sample rate and length of the time series, resulting in challenges and uncertainties in interpreting results. In this study, we examine the effects of different parameters on the distribution of the trend obtained from Kendall's τ, and provide insights into how to choose these parameters. We also suggest the use of the non-parametric Mann-Kendall test to evaluate the significance of a Kendall's τ value. The non-parametric test is computationally much faster compared with the traditional parametric auto-regressive, moving-average model test.

Deep learning for centre manifold reduction and stability analysis in nonlinear systems.

Bifurcations cause large qualitative and quantitative changes in the dynamics of nonlinear systems with slowly varying parameters. These changes most often are due to modifications that occur in a low-dimensional subspace of the overall system dynamics. The key challenge is to determine what that low-dimensional subspace is, and construct a low-order model that governs the dynamics in that subspace. Centre manifold theory can provide a theoretical means to construct such low-order models for strongly nonlinear systems that undergo bifurcations. Performing a centre manifold analysis, however, is particularly challenging when the system dimensionality is high or impossible when an accurate model of the system is not available. This paper introduces a data-driven approach for identifying a reduced order model of the system based on centre manifold theory. The approach does not require a model of the full order system. Instead, a deep learning approach capable of identifying the centre manifold and the transformation to the centre space is created using measurements of the system dynamics from random perturbations. This approach unravels the characteristics of the system dynamics in the vicinity of bifurcations, providing critical information regarding the behaviour of the system. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Data-driven prediction in dynamical systems: recent developments.

In recent years, we have witnessed a significant shift toward ever-more complex and ever-larger-scale systems in the majority of the grand societal challenges tackled in applied sciences. The need to comprehend and predict the dynamics of complex systems have spurred developments in large-scale simulations and a multitude of methods across several disciplines. The goals of understanding and prediction in complex dynamical systems, however, have been hindered by high dimensionality, complexity and chaotic behaviours. Recent advances in data-driven techniques and machine-learning approaches have revolutionized how we model and analyse complex systems. The integration of these techniques with dynamical systems theory opens up opportunities to tackle previously unattainable challenges in modelling and prediction of dynamical systems. While data-driven prediction methods have made great strides in recent years, it is still necessary to develop new techniques to improve their applicability to a wider range of complex systems in science and engineering. This focus issue shares recent developments in the field of complex dynamical systems with emphasis on data-driven, data-assisted and artificial intelligence-based discovery of dynamical systems. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Mathematical model of the feedback between global supply chain disruption and COVID-19 dynamics.

The pandemic of COVID-19 has become one of the greatest threats to human health, causing severe disruptions in the global supply chain, and compromising health care delivery worldwide. Although government authorities sought to contain the spread of SARS-CoV-2, by restricting travel and in-person activities, failure to deploy time-sensitive strategies in ramping-up of critical resource production exacerbated the outbreak. Here, we developed a mathematical model to analyze the effects of the interaction between supply chain disruption and infectious disease dynamics using coupled production and disease networks built on global data. Analysis of the supply chain model suggests that time-sensitive containment strategies could be created to balance objectives in pandemic control and economic losses, leading to a spatiotemporal separation of infection peaks that alleviates the societal impact of the disease. A lean resource allocation strategy can reduce the impact of supply chain shortages from 11.91 to 1.11% in North America. Our model highlights the importance of cross-sectoral coordination and region-wise collaboration to optimally contain a pandemic and provides a framework that could advance the containment and model-based decision making for future pandemics.

Data-driven identification of reliable sensor species to predict regime shifts in ecological networks.

Signals of critical slowing down are useful for predicting impending transitions in ecosystems. However, in a system with complex interacting components not all components provide the same quality of information to detect system-wide transitions. Identifying the best indicator species in complex ecosystems is a challenging task when a model of the system is not available. In this paper, we propose a data-driven approach to rank the elements of a spatially distributed ecosystem based on their reliability in providing early-warning signals of critical transitions. The proposed method is rooted in experimental modal analysis techniques traditionally used to identify structural dynamical systems. We show that one could use natural system fluctuations and the system responses to small perturbations to reveal the slowest direction of the system dynamics and identify indicator regions that are best suited for detecting abrupt transitions in a network of interacting components. The approach is applied to several ecosystems to demonstrate how it successfully ranks regions based on their reliability to provide early-warning signals of regime shifts. The significance of identifying the indicator species and the challenges associated with ranking nodes in networks of interacting components are also discussed.

Collective intracellular cargo transport by multiple kinesins on multiple microtubules.

The transport of intracellular organelles is accomplished by groups of molecular motors, such as kinesin, myosin, and dynein. Previous studies have demonstrated that the cooperation between kinesins on a track is beneficial for long transport. However, within crowded three-dimensional (3D) cytoskeletal networks, surplus motors could impair transport and lead to traffic jams of cargos. Comprehensive understanding of the effects of the interactions among molecular motors, cargo, and tracks on the 3D cargo transport dynamics is still lack. In this work, a 3D stochastic multiphysics model is introduced to study the synergistic and antagonistic motions of kinesin motors walking on multiple mircotubules (MTs). Based on the model, we show that kinesins attaching to a common cargo can interact mechanically through the transient forces in their cargo linkers. Under different environmental conditions, such as different MT topologies and kinesin concentrations, the transient forces in the kinesins, the stepping frequency and the binding and unbinding probabilities of kinesins are changed substantially. Therefore, the macroscopic transport properties, specifically the stall force of the cargo, the transport direction at track intersections, and the mean-square displacement (MSD) of the cargo along the MT bundles vary over the environmental conditions. In general, conditions that improve the synergistic motion of kinesins increase the stall force of the cargo and the capability of maintaining the transport. In contrast, the antagonistic motion of kinesins temporarily traps the cargo and slows down the transport. Furthermore, this study predicts an optimal number of kinesins for the cargo transport at MT intersections and along MT bundles.

Application of explainable ensemble artificial intelligence model to categorization of hemodialysis-patient and treatment using nationwide-real-world data in Japan.

BACKGROUND: Although dialysis patients are at a high risk of death, it is difficult for medical practitioners to simultaneously evaluate many inter-related risk factors. In this study, we evaluated the characteristics of hemodialysis patients using machine learning model, and its usefulness for screening hemodialysis patients at a high risk of one-year death using the nation-wide database of the Japanese Society for Dialysis Therapy. MATERIALS AND METHODS: The patients were separated into two datasets (n = 39,930, 39,930, respectively). We categorized hemodialysis patients in Japan into new clusters generated by the K-means clustering method using the development dataset. The association between a cluster and the risk of death was evaluated using multivariate Cox proportional hazards models. Then, we developed an ensemble model composed of the clusters and support vector machine models in the model development phase, and compared the accuracy of the prediction of mortality between the machine learning models in the model validation phase. RESULTS: Average age of the subjects was 65.7±12.2 years; 32.7% had diabetes mellitus. The five clusters clearly distinguished the groups on the basis of their characteristics: Cluster 1, young male, and chronic glomerulonephritis; Cluster 2, female, and chronic glomerulonephritis; Cluster 3, diabetes mellitus; Cluster 4, elderly and nephrosclerosis; Cluster 5, elderly and protein energy wasting. These clusters were associated with the risk of death; Cluster 5 compared with Cluster 1, hazard ratio 8.86 (95% CI 7.68, 10.21). The accuracy of the ensemble model for the prediction of 1-year death was 0.948 and higher than those of logistic regression model (0.938), support vector machine model (0.937), and deep learning model (0.936). CONCLUSIONS: The clusters clearly categorized patient on their characteristics, and reflected their prognosis. Our real-world-data-based machine learning system is applicable to identifying high-risk hemodialysis patients in clinical settings, and has a strong potential to guide treatments and improve their prognosis.

Effect of Multi-Frequency Whole-Body Vibration on Muscle Activation, Metabolic Cost and Regional Tissue Oxygenation.

Prolonged immobilization from a critical illness can result in significant muscle atrophy. Whole-body vibration (WBV) could potentially attenuate the issue of muscle atrophy; however, there exists no device that could potentially provide WBV in supine position that is suitable for critically ill patients. Hence, the purpose of this study was to develop a new wearable suit, called therapeutic vibration device (TVD), that can provide WBV in supine position and test its effects on physiologic markers of physical activity including muscle activation, oxygen consumption (VO2), and regional hemoglobin oxygen saturation (rSO2). The prototype TVD delivered multi-frequency WBV axially to 19 healthy participants in supine position for 10 minutes simultaneously at 25 Hz/4.2 grms on the feet and 15 Hz/0.7 grms on the shoulders. Muscle activation was recorded by electromyography (EMG), VO2 was measured by indirect calorimetry and rSO2 was recorded by near-infrared spectroscopy. Recordings were collected from each participant from multiple body locations, on three separate days, at baseline and during the intervention. Acceleration was also recorded to gain insight into transmissibility and coherence. Repeated-measures ANOVA using Bonferroni correction revealed that the muscle activity significantly increased by 4% - 62% (p < 0.05), VO2 improved by 22.3% (p < 0.05) and rSO2 increased by 1.4% - 4.5% (p < 0.05) compared to baseline. WBV provided by the TVD is capable of producing physiologic responses consistent with mild physical activity. Such effects could potentially be valuable as an adjunct to physical therapy for early mobilization to prevent atrophy occurring from prolonged immobilization.

The statistics of epidemic transitions.

Emerging and re-emerging pathogens exhibit very complex dynamics, are hard to model and difficult to predict. Their dynamics might appear intractable. However, new statistical approaches-rooted in dynamical systems and the theory of stochastic processes-have yielded insight into the dynamics of emerging and re-emerging pathogens. We argue that these approaches may lead to new methods for predicting epidemics. This perspective views pathogen emergence and re-emergence as a "critical transition," and uses the concept of noisy dynamic bifurcation to understand the relationship between the system observables and the distance to this transition. Because the system dynamics exhibit characteristic fluctuations in response to perturbations for a system in the vicinity of a critical point, we propose this information may be harnessed to develop early warning signals. Specifically, the motion of perturbations slows as the system approaches the transition.

Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems.

Many ecological systems are subject critical transitions, which are abrupt changes to contrasting states triggered by small changes in some key component of the system. Temporal early warning signals such as the variance of a time series, and spatial early warning signals such as the spatial correlation in a snapshot of the system's state, have been proposed to forecast critical transitions. However, temporal early warning signals do not take the spatial pattern into account, and past spatial indicators only examine one snapshot at a time. In this study, we propose the use of eigenvalues of the covariance matrix of multiple time series as early warning signals. We first show theoretically why these indicators may increase as the system moves closer to the critical transition. Then, we apply the method to simulated data from several spatial ecological models to demonstrate the method's applicability. This method has the advantage that it takes into account only the fluctuations of the system about its equilibrium, thus eliminating the effects of any change in equilibrium values. The eigenvector associated with the largest eigenvalue of the covariance matrix is helpful for identifying the regions that are most vulnerable to the critical transition.

A mathematical and computational model of the calcium dynamics in Caenorhabditis elegans ASH sensory neuron.

We propose a mathematical and computational model that captures the stimulus-generated Ca2+ transients in the C. elegans ASH sensory neuron. The rationale is to develop a tool that will enable a cross-talk between modeling and experiments, using modeling results to guide targeted experimental efforts. The model is built based on biophysical events and molecular cascades known to unfold as part of neurons' Ca2+ homeostasis mechanism, as well as on Ca2+ signaling events. The state of ion channels is described by their probability of being activated or inactivated, and the remaining molecular states are based on biochemically defined kinetic equations or known biochemical motifs. We estimate the parameters of the model using experimental data of hyperosmotic stimulus-evoked Ca2+ transients detected with a FRET sensor in young and aged worms, unstressed and exposed to oxidative stress. We use a hybrid optimization method composed of a multi-objective genetic algorithm and nonlinear least-squares to estimate the model parameters. We first obtain the model parameters for young unstressed worms. Next, we use these values of the parameters as a starting point to identify the model parameters for stressed and aged worms. We show that the model, in combination with experimental data, corroborates literature results. In addition, we demonstrate that our model can be used to predict ASH response to complex combinations of stimulation pulses. The proposed model includes for the first time the ASH Ca2+ dynamics observed during both "on" and "off" responses. This mathematical and computational effort is the first to propose a dynamic model of the Ca2+ transients' mechanism in C. elegans neurons, based on biochemical pathways of the cell's Ca2+ homeostasis machinery. We believe that the proposed model can be used to further elucidate the Ca2+ dynamics of a key C. elegans neuron, to guide future experiments on C. elegans neurobiology, and to pave the way for the development of more mathematical models for neuronal Ca2+ dynamics.

Rate of recovery from perturbations as a means to forecast future stability of living systems.

Anticipating critical transitions in complex ecological and living systems is an important need because it is often difficult to restore a system to its pre-transition state once the transition occurs. Recent studies demonstrate that several indicators based on changes in ecological time series can indicate that the system is approaching an impending transition. An exciting question is, however, whether we can predict more characteristics of the future system stability using measurements taken away from the transition. We address this question by introducing a model-less forecasting method to forecast catastrophic transition of an experimental ecological system. The experiment is based on the dynamics of a yeast population, which is known to exhibit a catastrophic transition as the environment deteriorates. By measuring the system's response to perturbations prior to transition, we forecast the distance to the upcoming transition, the type of the transition (i.e., catastrophic/non-catastrophic) and the future equilibrium points within a range near the transition. Experimental results suggest a strong potential for practical applicability of this approach for ecological systems which are at risk of catastrophic transitions, where there is a pressing need for information about upcoming thresholds.

Dynamic model for kinesin-mediated long-range transport and its local traffic jam caused by tau proteins.

In neurons, several intracellular cargoes are transported by motor proteins (kinesins) which walk on microtubules (MTs). However, kinesins can possibly unbind from the MTs before they reach their destinations. The unbound kinesins randomly diffuse in neurons until they bind to MTs. Then, they walk again along the MTs to continue their tasks. Kinesins repeat this cycle of motion until they transport their cargoes to the destinations. However, most previous models mainly focused on the motion of kinesins when they walk on MTs. Thus, a new model is required to encompass the various types of kinesin motion. We developed a comprehensive model and studied the long-range axonal transport of neurons using the model. To enhance reliability of the model, it was constructed based on multiphysics on kinesin motion (i.e., chemical kinetics, diffusion, fluid dynamics, nonlinear dynamics, and stochastic characteristics). Also, parameter values for kinesin motions are carefully obtained by comparing the model predictions and several experimental observations. The axonal transport can be degraded when a large number of binding sites on MTs are blocked by excessive tau proteins. By considering the interference between walking kinesins and tau molecules on MTs, effects of tau proteins on the axonal transport are studied. One of the meaningful predictions obtained from the model is that the velocity is not an effective metric to estimate the degradation of the transport because the decrease in velocity is not noticeable when the concentration of tau protein is not high. However, our model shows that the transport locally changes near tau molecules on MTs even when the change in the velocity is not significant. Thus, a statistical method is proposed to detect this local change effectively. The advantage of this method is that a value obtained from this method is highly sensitive to the concentration of tau protein. Another benefit of this method is that this highly sensitive value can be acquired with relatively low precision and low temporal resolution considering the time scale and length scale of the kinesin motion. This method can be used to estimate the condition of the axonal transport system.

Effects of Obstacles on the Dynamics of Kinesins, Including Velocity and Run Length, Predicted by a Model of Two Dimensional Motion.

Kinesins are molecular motors which walk along microtubules by moving their heads to different binding sites. The motion of kinesin is realized by a conformational change in the structure of the kinesin molecule and by a diffusion of one of its two heads. In this study, a novel model is developed to account for the 2D diffusion of kinesin heads to several neighboring binding sites (near the surface of microtubules). To determine the direction of the next step of a kinesin molecule, this model considers the extension in the neck linkers of kinesin and the dynamic behavior of the coiled-coil structure of the kinesin neck. Also, the mechanical interference between kinesins and obstacles anchored on the microtubules is characterized. The model predicts that both the kinesin velocity and run length (i.e., the walking distance before detaching from the microtubule) are reduced by static obstacles. The run length is decreased more significantly by static obstacles than the velocity. Moreover, our model is able to predict the motion of kinesin when other (several) motors also move along the same microtubule. Furthermore, it suggests that the effect of mechanical interaction/interference between motors is much weaker than the effect of static obstacles. Our newly developed model can be used to address unanswered questions regarding degraded transport caused by the presence of excessive tau proteins on microtubules.

Forecasting Bifurcations from Large Perturbation Recoveries in Feedback Ecosystems.

Forecasting bifurcations such as critical transitions is an active research area of relevance to the management and preservation of ecological systems. In particular, anticipating the distance to critical transitions remains a challenge, together with predicting the state of the system after these transitions are breached. In this work, a new model-less method is presented that addresses both these issues based on monitoring recoveries from large perturbations. The approach uses data from recoveries of the system from at least two separate parameter values before the critical point, to predict both the bifurcation and the post-bifurcation dynamics. The proposed method is demonstrated, and its performance evaluated under different levels of measurement noise, with two ecological models that have been used extensively in previous studies of tipping points and alternative steady states. The first one considers the dynamics of vegetation under grazing; the second, those of macrophyte and phytoplankton in shallow lakes. Applications of the method to more complex situations are discussed together with the kinds of empirical data needed for its implementation.

Highly loaded behavior of kinesins increases the robustness of transport under high resisting loads.

Kinesins are nano-sized biological motors which walk by repeating a mechanochemical cycle. A single kinesin molecule is able to transport its cargo about 1 µm in the absence of external loads. However, kinesins perform much longer range transport in cells by working collectively. This long range of transport by a team of kinesins is surprising because the motion of the cargo in cells can be hindered by other particles. To reveal how the kinesins are able to accomplish their tasks of transport in harsh intracellular circumstances, stochastic studies on the kinesin motion are performed by considering the binding and unbinding of kinesins to microtubules and their dependence on the force acting on kinesin molecules. The unbinding probabilities corresponding to each mechanochemical state of kinesin are modeled. The statistical characterization of the instants and locations of binding are captured by computing the probability of unbound kinesin being at given locations. It is predicted that a group of kinesins has a more efficient transport than a single kinesin from the perspective of velocity and run length. Particularly, when large loads are applied, the leading kinesin remains bound to the microtubule for long time which increases the chances of the other kinesins to bind to the microtubule. To predict effects of this behavior of the leading kinesin under large loads on the collective transport, the motion of the cargo is studied when the cargo confronts obstacles. The result suggests that the behavior of kinesins under large loads prevents the early termination of the transport which can be caused by the interference with the static or moving obstacles.

Metrics for characterizing collective transport by multiple dimeric kinesins.

Kinesin is a processive molecular motor which transports various cellular cargos by converting chemical energy into mechanical movements. Although the motion of a single molecule has been characterized in several studies, the dynamics of collective transport remains controversial. Since the chemical reactions fueling molecular motors are stochastic processes, the movements of coupled motors are not perfectly synchronized. The goal of this study is to develop metrics to analyze the level of synchronization of coupled (stochastic) motors. The correlation among movements of coupled motors, the slackness, the cooperativity, and the power loss of kinesins are explored using the developed metrics. These metrics can be extended to characterize collective work done by other molecular motors also.

The effects of viscoelastic fluid on kinesin transport.

Kinesins are molecular motors which transport various cargoes in the cytoplasm of cells and are involved in cell division. Previous models for kinesins have only targeted their in vitro motion. Thus, their applicability is limited to kinesin moving in a fluid with low viscosity. However, highly viscoelastic fluids have considerable effects on the movement of kinesin. For example, the high viscosity modifies the relation between the load and the speed of kinesin. While the velocity of kinesin has a nonlinear dependence with respect to the load in environments with low viscosity, highly viscous forces change that behavior. Also, the elastic nature of the fluid changes the velocity of kinesin. The new mechanistic model described in this paper considers the viscoelasticity of the fluid using subdiffusion. The approach is based on a generalized Langevin equation and fractional Brownian motion. Results show that a single kinesin has a maximum velocity when the ratio between the viscosity and elasticity is about 0.5. Additionally, the new model is able to capture the transient dynamics, which allows the prediction of the motion of kinesin under time varying loads.