James Keener : Flexing your Protein muscles: How to Pull with a Burning Rope
- Mathematical Biology ( 727 Views )The segregation of chromosomes during cell division is accomplished by kinetochore machinery that uses depolymerizing microtubules to pull the chromosomes to opposite poles of the dividing cell. While much is known about molecular motors that pull by walking or push by polymerizing, the mechanism of how a pulling force can be achieved by depolymerization is still unresolved. In this talk, I will describe a new model for the depolymerization motor that is used by eukaryotic cells to segregate chromosomes during mitosis. In the process we will explore the use of Huxley-type models (population models) of protein binding and unbinding to study load-velocity curves of several different motor-like proteins.
Shweta Bansal : Got flu? Using small and big data to understand influenza transmission, surveillance and control
- Mathematical Biology ( 301 Views )Traditional infectious disease epidemiology is built on the foundation of high quality and high accuracy data on disease and behavior. While these data are usually characterized by smallsize, they benefit from designed sampling schemes that make it possible to make population-level inferences. On the other hand, digital infectious disease epidemiology uses existing digital traces, re-purposing them to identify patterns in health-related processes. In this talk, I will discuss our work using data from small epidemiological studies as well as administrative “big data” to understand influenza transmission dynamics and inform disease surveillance and control.
John Gemmer : Nature’s Forms are Frilly, Flexible and Functional
- Mathematical Biology ( 259 Views )Many patterns in Nature and industry arise from the system minimizing an appropriate energy. Torn plastic sheets and growing leaves provide striking examples of pattern forming systems which can transition from single wavelength geometries (leaves) to complex fractal-like shapes (lettuce). These fractal-like patterns seem to have many length scales, i.e. the same amount of extra detail can be seen when looking closer (“statistical self-similarity”). It is a mystery how such complex patterns could arise from energy minimization alone. In this talk I will address this puzzle by showing that such patterns naturally arise from the sheet adopting a hyperbolic non-Euclidean geometry. However, there are many different hyperbolic geometries that the growing leaf could select. I will show using techniques from analysis, differential geometry and numerical optimization that the fractal like patterns are indeed the natural minimizers for the system. I will also discuss the implications of our work to developing shape changing soft matter which can be implemented in soft machines.
Rachel Howard : Monitoring the systemic immune response to cancer therapy
- Mathematical Biology ( 246 Views )Complex interactions occur between tumor and host immune system during cancer development and treatment, and a weak systemic immune response can be prognostic of poor patient outcomes. We strive to not only better understand the dynamic behavior of circulating immune cell populations before and during cancer therapy, but also to monitor these dynamic changes to facilitate real-time prediction of patient outcomes and potentially therapy adaptation. I will provide examples of both theoretical (mathematical) and data-driven (epidemiological) approaches to incorporating established systemic immune markers into clinical decision-making. First, coupling models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy, in turn identifying the optimal treatment target for maximum reduction of global tumor burden. Second, improved understanding of how circulating immune markers vary both within and between individual patients can allow more accurate risk stratification at diagnosis, and personalized prediction of patient response to therapy. The importance of multi-disciplinary collaborations in making predictive and prognostic models clinically relevant will be discussed.
Daniel Lew : Modeling the effect of vesicle traffic on polarity establishment in yeast
- Mathematical Biology ( 231 Views )There are two generally accepted models for the cell biological positive feedback loops that allow yeast cells to break symmetry and establish an axis of polarity. Both have been subjects of published mathematical analyses. Here I will argue that the models used to support a vesicle trafficking model incorporated a simplifying assumption that seemed innocuous but in fact was critical to their success. The assumption is not physically plausible, and its removal means that the model fails. I will show how changing other assumptions can make the model work, but there is no experimental support for those changes. And without them, the vesicle trafficking model perturbs polarity, rather than establishing polarity
Veronica Ciocanel : Stochastic and continuum dynamics in cellular transport
- Mathematical Biology ( 227 Views )The cellular cytoskeleton ensures the dynamic transport, localization, and anchoring of various proteins and vesicles. In the development of egg cells into embryos, messenger RNA (mRNA) molecules bind and unbind to and from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of partial differential equations as well as stochastic processes and derive the effective velocity and diffusivity of cargo at large time for a general class of problems. Including the geometry of the microtubule filaments allows for better prediction of particle localization and for investigation of previously unexplored mechanisms. Our numerical studies incorporating model microtubule structures suggest that anchoring of mRNA-molecular motor complexes may be necessary in localization, to promote healthy development of oocytes into embryos.
Johannes Reiter : Minimal intratumoral heterogeneity in untreated cancers
- Mathematical Biology ( 219 Views )Genetic intratumoral heterogeneity is a natural consequence of imperfect DNA replication. Any two randomly selected cells, whether normal or cancerous, are therefore genetically different. I will discuss the extent of genetic heterogeneity within untreated cancers with particular regard to its clinical relevance. While genomic heterogeneity within primary tumors is associated with relapse, heterogeneity among treatment‑naïve metastases has not been comprehensively assessed. We analyzed sequencing data for 76 untreated metastases from 20 patients and inferred cancer phylogenies for breast, colorectal, endometrial, gastric, lung, melanoma, pancreatic, and prostate cancers. We found that within individual patients a large majority of driver gene mutations are common to all metastases. Further analysis revealed that the driver gene mutations that were not shared by all metastases are unlikely to have functional consequences. A mathematical model of tumor evolution and metastasis formation provides an explanation for the observed driver gene homogeneity. Last, we found that individual metastatic lesions responded concordantly to targeted therapies in 91% of 44 patients. These data indicate that the cells within the primary tumors that gave rise to metastases are genetically homogeneous with respect to functional driver gene mutations and suggest that future efforts to develop combination therapies have the capacity to be curative.
John Dallon : Modeling Fibroblast Populated Collagen Lattices
- Mathematical Biology ( 215 Views )In order to better understand wound contraction fibroblast populated collagen lattices have been studied for many years. In this talk I will discuss mathematical models for lattice contraction. The models are formulated with components at the cellular and sub cellular level with the goal of understanding the macroscopic behavior of the lattice.
Arthur Sherman : Diabetes Pathogenesis as a Threshold-Crossing Process
- Mathematical Biology ( 205 Views )It has long been accepted that type 1 diabetes results from a lack of insulin, as the insulin-secreting pancreatic beta cells are destroyed by an autoimmune process. In contrast, the cause of type 2 diabetes (T2D) is less clear. Most people with pre-diabetes or in the early stages of T2D have abnormally high plasma insulin concentrations, and insulin rises before glucose does. We show that these difficulties are resolved by a mathematical model in which the onset of T2D is represented by the crossing of a threshold. The threshold is atypical in some respects and requires consideration of the slow manifolds to avoid incorrect conclusions.
Grzegorz A. Rempala, PhD DSc : Contact Processes and Stochastic Models of Epidemics
- Mathematical Biology ( 204 Views )I will discuss some old and new results related to the analysis of stochastic SIR-type epidemics on a configuration model (CM) random graph having a fixed degree distribution p_k. In particular, I will describe the relevant large graph limit result which yields the law of large numbers (LLN) for the edge-based process. I will also discuss the applications of the LLN approximation in building a "network-free" SIR Markov hybrid model which can be used for epidemic parameters inference. The hybrid model idea appears particularly relevant in the context of the recent Ebola and the Zika epidemics.
Jeremy Gunawardena : The Hopfield Barrier in eukaryotic gene regulation
- Mathematical Biology ( 201 Views )John Hopfield pointed out, in his seminal paper on kinetic proofreading, that if a biochemical system operates at thermodynamic equilibrium there is a barrier to how well it can achieve high-fidelity in transcription and translation. Hopfield showed that the only way to bypass this barrier is to dissipate energy and maintain the system away from equilibrium. Eukaryotic gene regulation uses dissipative mechanisms, such as nucleosome remodelling, DNA methylation and post-translational modification of histones, which are known to play a critical regulatory role but have been largely ignored in quantitative treatments. I will describe joint work with my colleague Angela DePace in which we use the recently-developed, graph-theoretic linear framework to show that the sharpness with which a gene is turned on or off in response to an upstream transcription factor is limited if the regulatory system operates at equilibrium, even with arbitrary degrees of higher-order cooperativity. In contrast, if the regulatory system is maintained away from equilibrium, substantially higher degrees of sharpness can be achieved. We suggest that achieving sharpness in gene regulation exhibits a Hopfield Barrier, and uncover, along the way, a new interpretation for the ubiquitously used, but poorly justified, Hill function.
Cristan Tomasetti : Stem cell divisions, somatic mutations, cancer etiology, and cancer prevention
- Mathematical Biology ( 193 Views )Cancers are caused by mutations that may be inherited, induced by environmental factors, or result from DNA replication errors (R). We studied the relationship between the number of normal stem cell divisions and the risk of 17 cancer types in 69 countries throughout the world. The data revealed a strong correlation (median = 0.80) between cancer incidence and normal stem cell divisions in all countries, regardless of their environment. The major role of R mutations in cancer etiology was supported by an independent approach, based solely on cancer genome sequencing and epidemiological data, which suggested that R mutations are responsible for two-thirds of the mutations in human cancers. All of these results are consistent with epidemiological estimates of the fraction of cancers that can be prevented by changes in the environment. Moreover, they accentuate the importance of early detection and intervention to reduce deaths from the many cancers arising from unavoidable R mutations.
Elliot Cartee : Control-Theoretic Models of Environmental Crime
- Mathematical Biology ( 188 Views )We present two models of perpetrators' decision-making in extracting resources from a protected area. It is assumed that the authorities conduct surveillance to counter the extraction activities, and that perpetrators choose their post-extraction paths to balance the time/hardship of travel against the expected losses from a possible detection. In our first model, the authorities are assumed to use ground patrols and the protected resources are confiscated as soon as the extractor is observed with them. The perpetrators' path-planning is modeled using the optimal control of randomly-terminated process. In our second model, the authorities use aerial patrols, with the apprehension of perpetrators and confiscation of resources delayed until their exit from the protected area. In this case the path-planning is based on multi-objective dynamic programming. Our efficient numerical methods are illustrated on several examples with complicated geometry and terrain of protected areas, non-uniform distribution of protected resources, and spatially non-uniform detection rates due to aerial or ground patrols.
David Basanta : The ecology of cancer: mathematical modelling and clinical implications
- Mathematical Biology ( 179 Views )Decades of research in cancer have yielded scant results other than highlighting the need for new approaches that could go beyond the tried and tested molecular-based ones. Recent clinical studies show that tumour heterogeneity and selection, the ingredients of Darwinian evolution, can explain cancer progression towards malignancy as well as recurrence after treatment. In this talk I will describe mathematical and computational models that explore cancer evolutionary dynamics and that can explain how the interactions between the tumour with its environment (the tumour ecosystem) can yield a better understanding of cancer biology and lead to better and more efficacious treatments for cancer patients.
Stephan Huckemann : Statistical challenges in shape prediction of biomolecules
- Mathematical Biology ( 176 Views )The three-dimensional higher-order structure of biomolecules
determines their functionality. While assessing primary structure is
fairly easily accessible, reconstruction of higher order structure is
costly. It often requires elaborate correction of atomic clashes,
frequently not fully successful. Using RNA data, we describe a purely
statistical method, learning error correction, drawing power from a
two-scale approach. Our microscopic scale describes single suites by
dihedral angles of individual atom bonds; here, addressing the
challenge of torus principal component analysis (PCA) leads to a
fundamentally new approach to PCA building on principal nested spheres
by Jung et al. (2012). Based on an observed relationship with a
mesoscopic scale, landmarks describing several suites, we use Fréchet
means for angular shape and size-and-shape, correcting
within-suite-backbone-to-backbone clashes. We validate this method by
comparison to reconstructions obtained from simulations approximating
biophysical chemistry and illustrate its power by the RNA example of
SARS-CoV-2.
This is joint work with Benjamin Eltzner, Kanti V. Mardia and Henrik
Wiechers.
Literature:
Eltzner, B., Huckemann, S. F., Mardia, K. V. (2018):
Torus principal component analysis with applications to RNA
structure. Ann. Appl. Statist. 12(2), 1332?1359.
Jung, S., Dryden, I. L., Marron, J. S. (2012):
Analysis of principal nested spheres. Biometrika, 99 (3), 551-568
Mardia, K. V., Wiechers, H., Eltzner, B., Huckemann, S. F. (2022).
Principal component analysis and clustering on manifolds. Journal of
Multivariate Analysis, 188, 104862,
https://www.sciencedirect.com/science/article/pii/S0047259X21001408
Wiechers, H., Eltzner, B., Mardia, K. V., Huckemann, S. F. (2021).
Learning torus PCA based classification for multiscale RNA backbone
structure correction with application to SARS-CoV-2. To appear in the
Journal of the Royal Statistical Society, Series C,
bioRxiv https://doi.org/10.1101/2021.08.06.455406
Stanca Ciupe : Models of antibody responses in HIV
- Mathematical Biology ( 174 Views )One of the first immunologic responses against HIV infection is the presence of neutralizing antibodies that seem able to inactivate several HIV strains. Moreover, in vitro studies have shown the existence of monoclonal antibodies that exhibit broad crossclade neutralizing potential. Yet their number is low and slow to develop in vivo. In this paper, we investigate the potential benefits of inducing poly-specific neutralizing antibodies in vivo throughout immunization. We develop a mathematical model that considers the activation of families of B lymphocytes producing poly-specific and strain-specific antibodies and use it to demonstrate that, even if such families are successful in producing neutralizing antibodies, competition between them may limit the poly-specific response allowing the virus to escape. We modify this model to account for viral evolution under the pressure of antibody responses in natural HIV infection as well as the need to neutralize more than one viral spike. The model can reproduce viral escape under certain conditions of B lymphocyte competition. Using these models we provide explanations for the observed antibody failure in controlling natural infection and predict quantitative measures that need to be satisfied for long-term control of HIV infection.
Rick Durrett : Spatial evolutionary games with weak selection
- Mathematical Biology ( 165 Views )Recently a mathematical theory has been developed for spatial games with weak selection, i.e., the payoff differences between strategies are small. The key to the analysis is that when space an time are suitably rescaled the limit is partial differential equation (PDE). This approach can be used to analyze all 2 x 2 games, but there are a number of 3 x 3 games for which the behavior of the limiting PDE is not known. In this talk we will describe simulation results for two cases that are not considered by rigorous results: rock-paper scissors and bistable games. We will begin by describing results for a two strategy game that arises from studying pancreatic cancer and shows that theoretical predictions work even when selection is not very weak. This is joint work with Mridu Nanda, a student at North Carolina School for Science and Math.
Michael Mackey : Understanding, treating and avoiding hematological disease
- Mathematical Biology ( 158 Views )This talk will trace many years of work mathematical modeling hematological diseases. The understand part talks about the use of mathematical to figure out what causes cyclical neutropenia, and the treat part refers to work on treating cyclical neutropenia using recombinant cytokines. The avoid part deals with current ongoing work trying to obviate the deleterious effects of chemotherapy on blood cell productionone of the major negative side effects of chemotherapy.
John Bush : Biocapillarity
- Mathematical Biology ( 157 Views )We report the results of our integrated experimental and theoretical investigations of biological systems dominated by interfacial effects. Particular attention is given to elucidating natural strategies for water-repellency, walking on water, underwater breathing, and drinking.
Nick Moore : How focused flexibility maximizes the thrust production of flapping wings
- Mathematical Biology ( 154 Views )Birds, insects, and fish all exploit the fact that flexible wings or fins generally perform better than their rigid counterparts. Given the task of designing an optimal wing, though, it is not clear how to best distribute the flexibility: Should the wing be uniformly flexible along its length, or could some advantage be gained by making certain sections more rigid than others? I will discuss this question by using a 2D small-amplitude model for the fluid-structure interaction combined with an efficient Chebyshev PDE solver. Numerical optimization shows that concentrating flexibility near the leading edge of the wing maximizes thrust production, an arrangement that resembles the torsional-joint flexibility mechanism found in insect wings. I will discuss the possibility of extending into three dimensions to address the question of optimal wing architecture more generally.
Adriana Dawes : Protein localization at the single cell level: Interplay between biochemistry, mechanics and geometry
- Mathematical Biology ( 153 Views )Cells are highly organized and complex structures, with the capacity to segregate specific factors to spatially disjoint regions in a process called polarization. Polarization, which specifies a spatial axis in the cell, is a highly conserved biological process and is required for proper embryonic development, wound healing, and many other normal and pathological biological functions. Despite the importance of polarization, we do not fully understand how this protein segregation is initiated and maintained. In this talk, I will show how we can use numerical and analytical approaches to investigate how symmetry breaking begins the process of polarization, and how the geometry of the cell may play a role in the establishment and maintenance of spatial patterns associated with polarization.
Jim Nolen : Sticky limit theorems for statistics in singular spaces.
- Mathematical Biology ( 147 Views )This talk is about extending classical limit theorems of probability (law of large numbers, central limit theorem) to a non-Euclidean setting. I'll talk about new and interesting phenomena observed when sampling independent points from certain singular geometric spaces. The main result is a limit theorem -- the "sticky central limit theorem" -- which applies to the mean or barycenter of a family of independent samples as the number of samples grows. The theorem shows that the geometry of the underlying space may have an interesting effect on the asymptotic fluctuations of the sample means, in a way that does not occur with independent samples in Euclidean space. One motivation for thinking about statistics in singular geometric spaces comes from evolutionary biology; one can consider phylogenetic trees as points in a metric space of the sort discussed in this talk. Apart from this basic motivation, however, the talk will have little biological content and will be mainly about probability.
Dan Forger : From a model network of 10,000 neurons to a smartphone app with >150,000 users: novel approaches to study daily timekeeping
- Mathematical Biology ( 145 Views )I will briefly describe mathematical models of networks of neurons and chemical reactions within neurons that generate daily (circadian) timekeeping. The numerical and analytical challenges of these models as well as the benefits in terms of biological predications will be highlighted. I will then explain how models can be used to find schedules that decrease the time needed to adjust to a new timezone by a factor of 2 or more. These optimal schedules have been implemented into a smartphone app, ENTRAIN, which collects data from users and in return helps them avoid jet-lag. We will use the data from this app to determine how the world sleeps. This presents a new paradigm in mathematical biology research where large-scale computing bridges the gap between basic mechanisms and human behavior and yields hypotheses that can be rapidly tested using mobile technology.
Steven Baer : Multiscale Modeling of Neural Subcircuits and Feedback Mechanisms in the Outer Plexiform Layer of the Retina
- Mathematical Biology ( 143 Views )Visual processing begins in the outer plexiform layer of the retina, where
bipolar, horizontal, and photoreceptor cells interact. In vertebrates, the
onset of dim backgrounds can enhance small spot flicker responses of
retinal horizontal cells. This flicker response is called background-
induced flicker enhancement. The underlying mechanism for the feedback
is unclear but competing hypotheses have been proposed. One is the GABA
hypothesis, which states that the inhibitory neurotransmitter GABA,
released from horizontal cells, mediates the feedback by blocking calcium
channels. Another is the ephaptic hypothesis, which contends that calcium
entry is regulated by changes in the electrical potential within the
intersynaptic space between cones and horizontal cells. In this study, a
continuum spine model of cone-horizontal cell synaptic circuitry is
formulated. The model captures two spatial scales - the scale of an
individual synapse and the scale of the receptive field involving hundreds
to thousands of synapses. We show that the ephaptic mechanism produces
reasonable qualitative agreement with the temporal dynamics exhibited by
flicker enhancement experiments. We find that although GABA produces
enhancement, this mechanism alone is insufficient to reproduce the
experimental results. We view this multiscale continuum approach as a
first step in formulating a multi-layer mathematical model of retinal
circuitry, which would include the other brain nuclei within the retina:
the inner plexiform layer where bipolar, amacrine, interplexiform, and
ganglion cells interact.
Mainak Patel : Temporal Binding Emerges as a Rapid and Accurate Encoding Tool Within a Network Model of the Locust Antennal Lobe
- Mathematical Biology ( 143 Views )The locust olfactory system interfaces with the external world through antennal receptor neurons (ORNs), which represent odors in a distributed, combinatorial manner. ORN axons bundle together to form the antennal nerve, which relays sensory information centrally to the antennal lobe (AL). Within the AL, an odor produces a stimulus-specific temporal progression of neuronal spiking, inspiring the hypothesis that the AL encodes odors through dynamically evolving ensembles of active cells. Such a coding strategy, however, requires higher olfactory centers to integrate a prolonged dynamic profile of AL signals prior to stimulus assessment, a process that is likely to be slow and inconsistent with the generation of quick behavioral responses. Our modeling work has led us to propose an alternate hypothesis: the dynamical interplay of fast and slow inhibition within the locust AL induces transient correlations in the spiking activity of an odor-dependent neural subset, giving rise to a temporal binding code and allowing rapid stimulus detection by downstream elements.
Phil Holmes : The neural dynamics of decision making: multiple scales in a single brain
- Mathematical Biology ( 141 Views )I will describe a range of models, from the cellular to cortical scales, that illuminate how we perceive stimuli and make decisions. Large networks composed of individual spiking neurons can capture biophysical details of neuromodulation and synaptic transmission, but their complexity renders them opaque to analysis. Employing methods of mean field and dynamical systems theory, I will argue that these high-dimensional stochastic differential equations can be reduced to simple drift-diffusion processes used by cognitive psychologists to fit behavioral data. This allows us to relate them to optimal methods from statistical decision theory, and prompts new questions on why we fail to make good choices.
Jake Taylor-King : Generalized Jump Processes and Osteocyte Network Formation
- Mathematical Biology ( 141 Views )My talk will have two parts. PART I, From Birds to Bacteria: Generalised Velocity Jump Processes. There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long distance movement state and a resting or local movement state. In this talk, I will give a mathematical description of this process, adapted from Friedrich et. al. (2006). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are tumultuous and therefore it is necessary to both simplify and derive summary statistics. We derive an expression for the mean squared displacement, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally a large time diffusive approximation is considered via a Cattaneo approximation (Hillen, 2004). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution. We also consider the Levy regime where the variance of the running distribution tends to infinity. This leads to a fractional diffusion equation for superdiffusive Levy walks and can be solved analytically. Our theory opens up new perspectives both for the systematic derivation of such equations, and for experimental data analysis of intermittent motion. I will also briefly discuss recent developments (by other researchers) within the field of velocity jump processes. PART II: Modelling Osteocyte Network Formation: Healthy and Cancerous Environments. Advanced prostate, breast, and lung cancer can metastasize to bone. In pathological bone, the highly regulated bone remodeling signaling pathway is disrupted. Within bone dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone formation. In the presence of a cancerous microenvironment, the morphology of this network changes. Commonly osteocytes appear to be either overdifferentiated (i.e., there are more dendrites) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, preliminary studies measuring the number of osteocytes per unit area using pathology slides show that the number density of osteocytes change from healthy to metastatic prostate and breast cancer xenografted mice. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and number density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We consider variations of our model to test specific physiological hypotheses related to osteoblast differentiation; for example we can predict how changing measurable biological parameters, such as rates of bone secretion, rates of dendrite growth and rates of osteoblast differentiation can allow for qualitatively different network morphologies, and vice versa. We thenuse our model to hypothesize reasons for the limited efficacy of zoledronate therapy on metastatic breast cancer.
Cecilia Clementi : A multiscale approach to characterize macromolecular dynamics and functions
- Mathematical Biology ( 140 Views )A multiscale approach to characterize macromolecular dynamics and functions The understanding of emerging collective behaviors in biomolecular complexes represents a major challenge in modern biophysics. As a first step toward the study of such processes we have applied multi-resolution nonlinear dimensionality reduction and diffusion analysis to obtain reliable low-dimensional representations and models for the dynamics of apparently high-dimensional complex systems such as proteins in a biological environment. The results clearly show that the proposed methods can efficiently find low dimensional representations of a complex process such as protein folding.