Mathematical modeling in chronobiology.

Circadian clocks are autonomous oscillators entrained by external Zeitgebers such as light-dark and temperature cycles. On the cellular level, rhythms are generated by negative transcriptional feedback loops. In mammals, the suprachiasmatic nucleus (SCN) in the anterior part of the hypothalamus plays the role of the central circadian pacemaker. Coupling between individual neurons in the SCN leads to precise self-sustained oscillations even in the absence of external signals. These neuronal rhythms orchestrate the phasing of circadian oscillations in peripheral organs. Altogether, the mammalian circadian system can be regarded as a network of coupled oscillators. In order to understand the dynamic complexity of these rhythms, mathematical models successfully complement experimental investigations. Here we discuss basic ideas of modeling on three different levels (1) rhythm generation in single cells by delayed negative feedbacks, (2) synchronization of cells via external stimuli or cell-cell coupling, and (3) optimization of chronotherapy.

Influence of wood ash pre-treatment on leaching behaviour, liming and fertilising potential.

In Denmark, increasing amounts of woody biomass are being used for the production of renewable energy, resulting in more wood ashes being generated. While these materials have been mainly landfilled, wood ashes may also be utilised for fertilizing and liming purposes on top of soils. Pre-treatments involving hardening or granulation may be carried out prior to soil application. In this study, two Danish wood ash samples were hardened and/or granulated. Lab-hardening induced rapid changes in the shape of the acid neutralisation capacity curve of the ashes. Up-flow column tests, assuming local equilibrium conditions, were employed to investigate the leaching from pre-treated ashes. Granules and loose ashes demonstrated similar leaching behaviours, indicating that similar geochemical processes were governing their leaching. In comparison with untreated fresh ashes, the hardened ashes demonstrated reduced leaching of Ca, Ba, Pb and Zn with concentration levels generally below or close to the analytical limits of quantification; to the contrary, the leaching of As, P, Sb, Si, V and Mg was enhanced in the hardened ashes. The release of alkalinity was reduced by hardening. In general, all granules were barely breakable by finger-pinching and they could withstand one month of continuous leaching, preserving their overall shape. The solubility of phosphorous in neutral ammonium citrate indicated that about 30-51% of the total P content in the ash samples was released, suggesting that the ashes could be potentially valuable as P-fertiliser if applied onto soil.

Oncogenic HRAS suppresses clusterin expression through promoter hypermethylation.

Silencing of gene expression by methylation of CpG islands in regulatory elements is frequently observed in cancer. However, an influence of the most common oncogenic signalling pathways onto DNA methylation has not yet been investigated thoroughly. To address this issue, we identified genes suppressed in HRAS-transformed rat fibroblasts but upregulated after treatment with the demethylating agent 5-Aza-2-deoxycytidine and with the MEK1,2 inhibitor U0126. Analysis of gene expression by microarray and Northern blot analysis revealed the MEK/ERK target genes clusterin, matrix metalloproteinase 2 (Mmp2), peptidylpropyl isomerase C-associated protein, syndecan 4, Timp2 and Thbs1 to be repressed in the HRAS-transformed FE-8 cells in a MEK/ERK- and methylation-dependent manner. Hypermethylation of putative regulatory elements in HRAS-transformed cells as compared to immortalized fibroblasts was detected within a CpG island 14.5 kb upstream of clusterin, within the clusterin promoter and within a CpG island of the Mmp2 promoter by bisulphite sequencing. Furthermore, hypermethylation of the clusterin promoter was observed 10 days after induction of HRAS in immortalized rat fibroblasts and a clear correlation between reduced clusterin expression and hypermethlyation could also be observed in distinct rat tissues. These results suggest that silencing of individual genes by DNA methylation is controlled by oncogenic signalling pathways, yet the mechanisms responsible for initial target gene suppression are variable.

Normalization strategies for cDNA microarrays.

Multiple Arabidopsis thaliana clones from an experimental series of cDNA microarrays are evaluated in order to identify essential sources of noise in the spotting and hybridization process. Theoretical and experimental strategies for an improved quantitative evaluation of cDNA microarrays are proposed and tested on a series of differently diluted control clones. Several sources of noise are identified from the data. Systematic and stochastic fluctuations in the spotting process are reduced by control spots and statistical techniques. The reliability of slide to slide comparison is critically assessed within the statistical framework of pattern matching and classification.

Sequence periodicity in complete genomes of archaea suggests positive supercoiling.

The topological state of genomic DNA is of importance for its replication, recombination and transcription. The wrapping of the DNA around nucleosomes is associated with sequence periodicities (Trifonov and Sussman, Proc. Natl. Acad. Sci. USA, 77, pp. 3816-20). Recently, also the negative supercoiling of eubacterial DNA was related to 11 base pair (bp) periodicity (Herzel et al. Physica A, 249, pp. 449-59). Archaeal plasmids and a virus-like particle from Sulfolobus are positively supercoiled, but the superhelical conformation of archaeal genomic DNA is still uncertain. The problem of superhelicity can now be addressed via a comparative statistical analysis of the available complete genomes. For this purpose one has to look for periodicities which are in phase with the helical repeat of 10-11 bp. Similar periodicities are induced, however, by the amphipatic character of alpha-helices of encoded proteins (Zhurkin, Nucl. Acids Res., 9, pp. 1963-71). We show that these protein-induced periodicities are extended over a few periods only. The periods of additional long-ranging oscillations deviate significantly from the value for free DNA. A period of 11 bp in Eubacteria reflects negative supercoiling, whereas the significantly different period of thermophilic Archaea close to 10 bp suggests positive supercoiling of archaeal genomes.

Statistical analysis of the DNA sequence of human chromosome 22.

We study statistical patterns in the DNA sequence of human chromosome 22, the first completely sequenced human chromosome. We find that (i). the 33.4 x 10(6) nucleotide long human chromosome exhibits long-range power-law correlations over more than four orders of magnitude, (ii). the entropies H(n) of the frequency distribution of oligonucleotides of length n (n-mers) grow sublinearly with increasing n, indicating the presence of higher-order correlations for all of the studied lengths 1

The modular structure of informational sequences.

It is shown that DNA sequences can be decomposed into smaller units much the same as texts can be decomposed into syllables, words, or groups of words. Those smaller units (modules) are extracted from DNA sequences according to statistical criteria. Tests with sequences of known modular structure (two novels and a FORTRAN source code) were performed. The rate to which DNA sequences can be decomposed into modules (modularity) turns out to be a very sensitive measure to distinguish DNA sequences from random sequences.

Bifurcations in excised larynx experiments.

Bifurcation analysis was applied to vocal fold vibration in excised larynx experiments. Phonation onset and vocal instabilities were studied in a parameter plane spanned by subglottal pressure and asymmetry of either vocal fold adduction or elongation. Various phonatory regimes were observed, including single vocal fold oscillations. Selected spectra demonstrated correspondence between these regimes and vocal registers noted in the literature. To illustrate the regions spanned by the various phonatory regimes, two-dimensional bifurcation diagrams were generated. Many instabilities or bifurcations were noted in the regions of coexistence, i.e., regions in which the phonatory regimes overlap. Bifurcations were illustrated with spectrograms and fundamental frequency contours. Where possible, results from these studies were related to clinical observations.

Genetic redundancy strengthens the circadian clock leading to a narrow entrainment range.

Circadian clocks are internal timekeepers present in almost all organisms. Driven by a genetic network of highly conserved structure, they generate self-sustained oscillations that entrain to periodic external signals such as the 24 h light-dark cycle. Vertebrates possess multiple, functionally overlapping homologues of the core clock genes. Furthermore, vertebrate clocks entrain to a range of periods three times as narrow as that of other organisms. We asked whether genetic redundancies play a role in governing entrainment properties and analysed locomotor activity rhythms of genetically modified mice lacking one set of clock homologues. Exposing them to non-24 h light-dark cycles, we found that the mutant mice have a wider entrainment range than the wild types. Spectral analysis furthermore revealed nonlinear phenomena of periodically forced self-sustained oscillators for which the entrainment range relates inversely to oscillator amplitude. Using the forced oscillator model to explain the observed differences in entrainment range between mutant and wild-type mice, we sought to quantify the overall oscillator amplitude of their clocks from the activity rhythms and found that mutant mice have weaker circadian clocks than wild types. Our results suggest that genetic redundancy strengthens the circadian clock leading to a narrow entrainment range in vertebrates.

Circadian transcription in liver.

Circadian rhythms regulate a wide range of cellular, physiological, metabolic and behavioral activities in mammals. The complexity of tissue- and day-time specific regulation of thousands of clock controlled genes (CCGs) suggests that many transcriptional regulators are involved. Our bioinformatic analysis is based on two published DNA-array studies from mouse liver. We search overrepresented transcription factor binding sites in promoter regions of CCGs using GC-matched controls. Analyzing a large set of CCG promoters, we find known motifs such as E-boxes, D-boxes and cAMP responsive elements. In addition, we find overrepresented GC-rich motifs (Sp1, ETF, Nrf1), AT-rich motifs (TBP, Fox04, MEF-2), Y-box motifs (NF-Y, C/EBP) and cell cycle regulators (E2F, Elk-1). In a subset of system-driven genes, we find overrepresented motifs of the serum response factor SRF and the estrogen receptor ER. The analysis of published ChIP data reveals that some of our predicted regulators (C/EBP, E2F, HNF-1, Myc, MEF-2) target relatively many clock controlled genes. Our analysis of CCG promoters contributes to an understanding of the complex transcriptional regulation of circadian rhythms in liver.

Phase response curves elucidating the dynamics of coupled oscillators.

Phase response curves (PRCs) are widely used in circadian clocks, neuroscience, and heart physiology. They quantify the response of an oscillator to pulse-like perturbations. Phase response curves provide valuable information on the properties of oscillators and their synchronization. This chapter discusses biological self-sustained oscillators (circadian clock, physiological rhythms, etc.) in the context of nonlinear dynamics theory. Coupled oscillators can synchronize with different frequency ratios, can generate toroidal dynamics (superposition of independent frequencies), and may lead to deterministic chaos. These nonlinear phenomena can be analyzed with the aid of a phase transition curve, which is intimately related to the phase response curve. For illustration purposes, this chapter discusses a model of circadian oscillations based on a delayed negative feedback. In a second part, the chapter provides a step-by-step recipe to measure phase response curves. It discusses specifications of this recipe for circadian rhythms, heart rhythms, neuronal spikes, central pattern generators, and insect communication. Finally, it stresses the predictive power of measured phase response curves. PRCs can be used to quantify the coupling strength of oscillations, to classify oscillator types, and to predict the complex dynamics of periodically driven oscillations.

SysBioMed report: advancing systems biology for medical applications.

The following report selects and summarises some of the conclusions and recommendations generated throughout a series of workshops and discussions that have lead to the publication of the Science Policy Briefing (SPB) Nr. 35, published by the European Science Foundation. (Large parts of the present text are directly based on the ESF SPB. Detailed recommendations with regard to specific application areas are not given here but can be found in the SPB. Issues related to mathematical modelling, including training and the need for an infrastructure supporting modelling are discussed in greater detail in the present text.)The numerous reports and publications about the advances within the rapidly growing field of systems biology have led to a plethora of alternative definitions for key concepts. Here, with 'mathematical modelling' the authors refer to the modelling and simulation of subcellular, cellular and macro-scale phenomena, using primarily methods from dynamical systems theory. The aim of such models is encoding and testing hypotheses about mechanisms underlying the functioning of cells. Typical examples are models for molecular networks, where the behaviour of cells is expressed in terms of quantitative changes in the levels of transcripts and gene products. Bioinformatics provides essential complementary tools, including procedures for pattern recognition, machine learning, statistical modelling (testing for differences, searching for associations and correlations) and secondary data extracted from databases.Dynamical systems theory is the natural language to investigate complex biological systems demonstrating nonlinear spatio-temporal behaviour. However, the generation of experimental data suitable to parameterise, calibrate and validate such models is often time consuming and expensive or not even possible with the technology available today. In our report, we use the term 'computational model' when mathematical models are complemented with information generated from bioinformatics resources. Hence, 'the model' is, in reality, an integrated collection of data and models from various (possibly heterogeneous) sources. The present report focuses on a selection of topics, which were identified as appropriate case studies for medical systems biology, and adopts a particular perspective which the authors consider important. We strongly believe that mathematical modelling represents a natural language with which to integrate data at various levels and, in doing so, to provide insight into complex diseases: 1. Modelling necessitates the statement of explicit hypotheses, a process which often enhances comprehension of the biological system and can uncover critical points where understanding is lacking. 2. Simulations can reveal hidden patterns and/or counter-intuitive mechanisms in complex systems. 3. Theoretical thinking and mathematical modelling constitute powerful tools to integrate and make sense of the biological and clinical information being generated and, more importantly, to generate new hypotheses that can then be tested in the laboratory.Medical Systems Biology projects carried out recently across Europe have revealed a need for action: 4. While the need for mathematical modelling and interdisciplinary collaborations is becoming widely recognised in the biological sciences, with substantial implications for the training and research funding mechanisms within this area, the medical sciences have yet to follow this lead. 5. To achieve major breakthroughs in Medical Systems Biology, existing academic funding schemes for large-scale projects need to be reconsidered. 6. The hesitant stance of the pharmaceutical industry towards major investment in systems biology research has to be addressed. 7. Leading medical journals should be encouraged to promote mathematical modelling.

Bifurcations in an asymmetric vocal-fold model.

A two-mass model of vocal-fold vibrations is analyzed with methods from nonlinear dynamics. Bifurcations are located in parameter planes of physiological interest (subglottal pressure, stiffness of the folds). It is shown that a sufficiently large tension imbalance of the left and right vocal fold induces bifurcations to subharmonic regimes, toroidal oscillations, and chaos. The corresponding attractors are characterized by phase portraits, spectra, and next-maximum maps. The relevance of these simulations for voice disorders such as laryngeal paralysis is discussed.

Whistle register and biphonation in a child's voice.

This paper deals with high-pitched vocalizations of a normal healthy child. In the transition region between whistle register and falsetto frequency jumps and coexisting pitches ('biphonation') are observed. The recorded signals are compared to simulations of an asymmetric two-mass model of vocal fold vibrations. As a possible mechanism of the whistle register vortex-induced vibrations are discussed.

Correlations in protein sequences and property codes.

Correlation functions in large sets of non-homologous protein sequences are analysed. Finite size corrections are applied and fluctuations are estimated. As symbol sequences have to be mapped to sequences of numbers to calculate correlation functions, several property codes are tested as such mappings. We found hydrophobicity autocorrelation functions to be strongly oscillating. Another strong signal is the monotonously decaying alpha-helix propensity autocorrelation function. Furthermore, we detected signals corresponding to an alteration of positively and negatively charged residues at a distance of 3-4 amino acids. To look beyond the property codes gained by the methods of physical chemistry, mappings yielding a strong correlation signal are sought for using a Monte Carlo simulation. The mappings leading to strong signals are found to be related to hydrophobicity of alpha-helix propensity. A cluster analysis of the top scoring mappings leads to two novel property codes. These two property codes are gained from sequence data only. They turn out to be similar to known property codes for hydrophobicity or polarity.

Effects of noise and inhomogeneous attractors in biochemical systems.

Biochemical systems capable of sustained oscillations and deterministic chaos were investigated, and effects of short-correlated noise on oscillations and chaos are reviewed. It is emphasized that fluctuations can be amplified drastically under certain circumstances and, therefore, they can be easily confused with truly chaotic behaviour. Biochemical systems exhibit typically inhomogeneous attractors. Inhomogeneity can be quantified by introducing local densities. The implication of inhomogeneity for time-series analysis is demonstrated by an example of experimental data from a chemical system.

Estimating the entropy of DNA sequences.

The Shannon entropy is a standard measure for the order state of symbol sequences, such as, for example, DNA sequences. In order to incorporate correlations between symbols, the entropy of n-mers (consecutive strands of n symbols) has to be determined. Here, an assay is presented to estimate such higher order entropies (block entropies) for DNA sequences when the actual number of observations is small compared with the number of possible outcomes. The n-mer probability distribution underlying the dynamical process is reconstructed using elementary statistical principles: The theorem of asymptotic equi-distribution and the Maximum Entropy Principle. Constraints are set to force the constructed distributions to adopt features which are characteristic for the real probability distribution. From the many solutions compatible with these constraints the one with the highest entropy is the most likely one according to the Maximum Entropy Principle. An algorithm performing this procedure is expounded. It is tested by applying it to various DNA model sequences whose exact entropies are known. Finally, results for a real DNA sequence, the complete genome of the Epstein Barr virus, are presented and compared with those of other information carriers (texts, computer source code, music). It seems as if DNA sequences possess much more freedom in the combination of the symbols of their alphabet than written language or computer source codes.