Susanna Röblitz

Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification

Given a row-stochastic matrix describing pairwise similarities between data objects, spectral clustering makes use of the eigenvectors of this matrix to perform dimensionality reduction for clustering in fewer dimensions. One example from this class of algorithms is the Robust Perron Cluster Analysis (PCCA+), which delivers a fuzzy clustering. Originally developed for clustering the state space of Markov chains, the method became popular as a versatile tool for general data classification problems. The robustness of PCCA+, however, cannot be explained by previous perturbation results, because the matrices in typical applications do not comply with the two main requirements: reversibility and nearly decomposability. We therefore demonstrate in this paper that PCCA+ always delivers an optimal fuzzy clustering for nearly uncoupled, not necessarily reversible, Markov chains with transition states.

A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions.

Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mechanistic mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. The model generates successive estrous cycles of 21 days, with three waves of follicle growth per cycle. The model contains 12 differential equations and 54 parameters. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included in the model are based on relations that in the literature have only been described qualitatively (i.e. stimulation and inhibition), the output of the model is surprisingly well in line with empirical data. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences.

A mathematical model of the human menstrual cycle for the administration of GnRH analogues

The paper presents a differential equation model for the feedback mechanisms between gonadotropin-releasing hormone (GnRH), follicle-stimulating hormone (FSH), luteinizing hormone (LH), development of follicles and corpus luteum, and the production of estradiol (E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual cycle. Compared to earlier human cycle models, there are three important differences: The model presented here (a) does not involve any delay equations, (b) is based on a deterministic modeling of the GnRH pulse pattern, and (c) contains less differential equations and less parameters. These differences allow for a faster simulation and parameter identification. The focus is on modeling GnRH-receptor binding, in particular, by inclusion of a pharmacokinetic/pharmacodynamic (PK/PD) model for a GnRH agonist, Nafarelin, and a GnRH antagonist, Cetrorelix, into the menstrual cycle model. The final mathematical model describes the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle of 12 healthy women. It correctly predicts hormonal changes following single and multiple dose administration of Nafarelin or Cetrorelix at different stages in the cycle.

Mechanisms regulating follicle wave patterns in the bovine estrous cycle investigated with a mathematical model

A normal bovine estrous cycle contains 2 or 3 waves of follicle development, and ovulation takes place in the last wave. However, the biological mechanisms that determine whether a cycle has 2 or 3 waves have not been elucidated. In a previous paper, we described a mathematical model of the bovine estrous cycle that generates cyclical fluctuations of hormones, follicles, and corpora lutea in estrous cycles of approximately 21 d for cows with a normal estrous cycle. The parameters in the model represent kinetic properties of the system with regard to synthesis, release, and clearance of hormones and growth and regression of follicles and corpora lutea. The initial model parameterization resulted in estrous cycles with 3 waves of follicular growth. Here, we use this model to explore which physiological mechanisms could affect the number of follicular waves. We hypothesized that some of the parameters related to follicle growth rate or to the time point of corpus luteum regression are likely candidates to affect the number of waves per cycle. We performed simulations with the model in which we varied the values of these parameters. We showed that variation of (combinations of) model parameters regulating follicle growth rate or time point of corpus luteum regression can change the model output from 3 to 2 waves of follicular growth in a cycle. In addition, alternating 2- and 3-wave cycles occurred. Some of the parameter changes seem to represent plausible biological mechanisms that could explain these follicular wave patterns. In conclusion, our simulations indicated likely parameters involved in the mechanisms that regulate the follicular wave pattern, and could thereby help to find causes of declined fertility in dairy cows.

Computing Expectation Values for Molecular Quantum Dynamics

We compute expectation values for the solution of the nuclear Schrodinger equation. The proposed particle method consists of three steps: sampling of the initial Wigner function, classical transport of the sampling points, and weighted phase space summation for the final computation of the expectation values. The Egorov theorem guarantees that the algorithm is second order accurate with respect to the semiclassical parameter. We present numerical experiments for a two-dimensional torsional potential with three different sets of initial data and for a six-dimensional Henon-Heiles potential. By construction, the computing times scale linearly with the number of initial sampling points and range between three seconds and one hour.

Advances in modeling of the bovine estrous cycle: Synchronization with PGF2α

Our model of the bovine estrous cycle is a set of ordinary differential equations which generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It describes the growth and decay of the follicles and the corpus luteum, as well as the change of the key reproductive hormones, enzymes and processes over time. In this work we describe recent developments of this model towards the administration of prostaglandin F2α. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after single dose administrations of synthetic prostaglandin F2α.

Lack of Associations between Female Hormone Levels and Visuospatial Working Memory, Divided Attention and Cognitive Bias across Two Consecutive Menstrual Cycles

Background: Interpretation of observational studies on associations between prefrontal cognitive functioning and hormone levels across the female menstrual cycle is complicated due to small sample sizes and poor replicability. Methods: This observational multisite study comprised data of n = 88 menstruating women from Hannover, Germany, and Zurich, Switzerland, assessed during a first cycle and n = 68 re-assessed during a second cycle to rule out practice effects and false-positive chance findings. We assessed visuospatial working memory, attention, cognitive bias and hormone levels at four consecutive time-points across both cycles. In addition to inter-individual differences we examined intra-individual change over time (i.e., within-subject effects). Results: Estrogen, progesterone and testosterone did not relate to inter-individual differences in cognitive functioning. There was a significant negative association between intra-individual change in progesterone and change in working memory from pre-ovulatory to mid-luteal phase during the first cycle, but that association did not replicate in the second cycle. Intra-individual change in testosterone related negatively to change in cognitive bias from menstrual to pre-ovulatory as well as from pre-ovulatory to mid-luteal phase in the first cycle, but these associations did not replicate in the second cycle. Conclusions: There is no consistent association between womens hormone levels, in particular estrogen and progesterone, and attention, working memory and cognitive bias. That is, anecdotal findings observed during the first cycle did not replicate in the second cycle, suggesting that these are false-positives attributable to random variation and systematic biases such as practice effects. Due to methodological limitations, positive findings in the published literature must be interpreted with reservation.

Solution of the chemical master equation by radial basis functions approximation with interface tracking

BackgroundThe chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.ResultsWe introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space.DiscussionIn order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.ConclusionsThe proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability.

Adaptive spectral clustering with application to tripeptide conformation analysis

A decomposition of a molecular conformational space into sets or functions (states) allows for a reduced description of the dynamical behavior in terms of transition probabilities between these states. Spectral clustering of the corresponding transition probability matrix can then reveal metastabilities. The more states are used for the decomposition, the smaller the risk to cover multiple conformations with one state, which would make these conformations indistinguishable. However, since the computational complexity of the clustering algorithm increases quadratically with the number of states, it is desirable to have as few states as possible. To balance these two contradictory goals, we present an algorithm for an adaptive decomposition of the position space starting from a very coarse decomposition. The algorithm is applied to small data classification problems where it was shown to be superior to commonly used algorithms, e.g., k-means. We also applied this algorithm to the conformation analysis of a tripeptide molecule where six-dimensional time series are successfully analyzed.

Linear Precision Glycomacromolecules with Varying Interligand Spacing and Linker Functionalities Binding to Concanavalin A and the Bacterial Lectin FimH

A series of precision glycomacromolecules is prepared following previously established solid phase synthesis allowing for controlled variations of interligand spacing and the overall number of carbohydrate ligands. In addition, now also different linkers are installed between the carbohydrate ligand and the macromolecular scaffold. The lectin binding behavior of these glycomacromolecules is then evaluated in isothermal titration calorimetry (ITC) and kinITC experiments using the lectin Concanavalin A (Con A) in its dimeric and tetrameric form. The results indicate that both sterical and statistical effects impact lectin binding of precision glycomacromolecules. Moreover, ITC results show that highest affinity toward Con A can be achieved with an ethyl phenyl linker, which parallels earlier findings with the bacterial lectin FimH. In this way, a first set of glycomacromolecule structures is selected for testing in a bacterial adhesion-inhibition study. Here, the findings point to a one-sugar binding mode mainly affected by sterical restraints of the nonbinding parts of the respective glycomacromolecule.