On some formulas in a partnership model from the perspective of a semi-Markov process

Many deterministic models of sexually transmitted diseases, as well as population models in general, contain elements of stochastic or statistical reasoning. An example of such a model is that of Dietz and Hadeler (1988) concerning sexually transmitted diseases in which there is partnership formation and dissolution. Among the interesting formulas in this paper, which enter into the analysis of the model, are those for the expected number of partners a male or female has during a lifetime. To a probabilist such formulas suggest the possibility that some stochastic process may be constructed so as to yield these formulas as well as others that may be of interest. The principal purpose of this paper is to demonstrate that such a stochastic process does indeed exist in the form of a three state semi-Markov process in continuous time with stationary laws of evolution and with a one-step density matrix determined by four parameters which were interpreted as constant latent risk functions in the classical theory of competing risks. This construction of a semi-Markov process not only provides a framework for the systematic derivation of the formulas of Dietz and Hadeler but also suggests pathways,for extensions to the age-dependent case.This research was partially supported by NATO Grant D.890350     

Karl-Peter Hadeler: His legacy in mathematical biology

Journal of Mathematical Biology - Karl-Peter Hadeler is a first-generation pioneer in mathematical biology. His work inspired the contributions to this special issue. In this preface we give a...     

High frequency spikes in long period blood cell oscillations

Several hematological diseases are characterised by oscillations of various blood cell populations. Two of these are a variant of chronic myelogenous leukemia (CML) and cyclical neutropenia (CN). These oscillations typically have long periods ranging from 20 to 60 days, despite the fact that the stem cell cycling time is thought to be of the order of 2–3 days. Clinical data from humans and laboratory data from the grey collie animal model of CN is suggestive of the idea that these long period oscillations may also contain higher frequency spiky oscillations. We show how such oscillations can be understood in the context of slow periodic stem cell oscillations, by analysing a two component differential-delay equation model of stem cell and neutrophil populations.For Karl Hadeler, on his 70th birthday, leader, teacher, colleague and friend.     

Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor

The asymptotic behavior of solutions of a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with two distributed delays and an external inhibitor is considered. The model presents a refinement of the one considered by Lu and Hadeler [Z. Lu, K.P. Hadeler, Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. Biosci. 167 (2000) p. 177]. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. Furthermore, it is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Conditions for boundedness of solutions and existence of non-negative equilibrium are given. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the uniform persistence of the system are also carried out. By constructing appropriate Liapunov-like functionals, some sufficient conditions of global attractivity to the extinction equilibria are obtained and the combined effects of the delays and the inhibitor are studied.     

An age-structured epidemic model of rotavirus with vaccination

The recent approval of a rotavirus vaccine in Mexico motivates this study on the potential impact of the use of such a vaccine on rotavirus prevention and control. An age-structured model that describes the rotavirus transmission dynamics of infections is introduced. Conditions that guarantee the local and global stability analysis of the disease-free steady state distribution as well as the existence of an endemic steady state distribution are established. The impact of maternal antibodies on the implementation of vaccine is evaluated. Model results are used to identify optimal age-dependent vaccination strategies. A convergent numerical scheme for the model is introduced but not implemented. This paper is dedicated to Prof. K. P. Hadeler, who continues to push the frontier of knowledge in mathematical biology.     

Comparison of microsatellites, single-nucleotide polymorphisms (SNPs) and composite markers derived from SNPs in linkage analysis

There is growing evidence that a map of dense single-nucleotide polymorphisms (SNPs) can outperform a map of sparse microsatellites for linkage analysis. There is also argument as to whether a clustered SNP map can outperform an evenly spaced SNP map. Using Genetic Analysis Workshop 14 simulated data, we compared for linkage analysis microsatellites, SNPs, and composite markers derived from SNPs. We encoded the composite markers in a two-step approach, in which the maximum identity length contrast method was employed to allow for recombination between loci. A SNP map 2.3 times as dense as a microsatellite map (approximately 2.9 cM compared to approximately 6.7 cM apart) provided slightly less information content (approximately 0.83 compared to approximately 0.89). Most inheritance information could be extracted when the SNPs were spaced < 1 cM apart. Comparing the linkage results on using SNPs or composite markers derived from them based on both 3 cM and 0.3 cM resolution maps, we showed that the inter-SNP distance should be kept small (< 1 cM), and that for multipoint linkage analysis the original markers and the derived composite markers had similar power; but for single point linkage analysis the resulting composite markers lead to more power. Considering all factors, such as information content, flexibility of analysis method, map errors, and genotyping errors, a map of clustered SNPs can be an efficient design for a genome-wide linkage scan.     

Cattaneo models for chemosensitive movement

 We derive models for chemosensitive movement based on Cattaneo's law of heat propagation with finite speed. We apply the model to pattern formation as observed in experiments with Dictyostelium discoideum, with Salmonella typhimurium and with Escherichia coli. For Salmonella typhimurium we make predictions on pattern formation which can be tested in experiments. We discuss the relations of the Cattaneo models to classical models and we develop an effective numerical scheme. Received: 8 October 2001 / Revised version: 2 August 2002 / Published online: 19 November 2002 Key words or phrases: Chemotaxis – Aggregation – Cattaneo model – Numerical schemes Acknowledgements. We are very grateful for comments of S. Noelle concerning the numerical scheme. We thank K.P. Hadeler and C. Schmeiser for helpful remarks. The research was supported by the Deutsche Forschungsgemeinschaft, research project ANumE and the Austrian Science Foundation, grant no. W008.     

iMap: a database-driven utility to integrate and access the genetic and physical maps of maize

MOTIVATION: Because of the unique biological features, a bioinformatic platform for the integrated genetic and physical map of maize is required for storing, integrating, accessing and visualizing the underlying data. RESULTS: The goal of the Maize Mapping Project is to develop a fully integrated genetic and physical map for maize. To display this integrated map, we have developed iMap. iMap has three main components: a relational database (iMapDB), a map graphic browser (iMap Viewer) and a search utility (iMap Search). iMapDB is populated with current genetic and physical map data, describing relationships among genetic loci, molecular markers and bacterial artificial chromosome (BAC) contigs. The database also contains integrated information produced by applying a set of anchoring rules to assign BAC contigs to specific locations on the genetic map. The iMap Viewer and iMap Search functions are combined in the user interface to allow viewing and retrieving many types of genetic and physical map data. The iMap Viewer features side-by-side chromosome-based displays of the genetic map and associated BAC contigs. For each genetic locus, information about marker type or contig can be viewed via pop-up windows that feature links to external data resources. Searches can be conducted for genetic locus, probe or sequence accession number; search results include relevant map positions, anchored BAC contigs and links to the graphical display of relevant chromosomes. iMap can be accessed at http://www.maizemap.org AVAILABILITY: The iMap utility package is available for non-commercial use upon request from the authors.     

Egocentric Representation Acquired from Offline Map Learning

It is widely accepted that people establish allocentric spatial representation after learning a map. However, it is unknown whether people can directly acquire egocentric representation after map learning. In two experiments, the participants learned a distal environment through a map and then performed the egocentric pointing tasks in that environment under three conditions: with the heading aligned with the learning perspective (baseline), after 240° rotation from the baseline (updating), and after disorientation (disorientation). Disorientation disrupted the internal consistency of pointing among objects when the participants learned the sequentially displayed map, on which only one object name was displayed at a time while the location of “self” remained on the screen all the time. However, disorientation did not affect the internal consistency of pointing among objects when the participants learned the simultaneously displayed map. These results suggest that the egocentric representation can be acquired from a sequentially presented map.     

Coexistence research requires more interdisciplinary communication

Coexistence theories develop rapidly at the ecology forefront, outpacing their experimental testing. I discuss the reasons for this gap, call on interdisciplinary researchers to construct a road map for coexistence research, and recommend the actions that should be implemented therein.

Coexistence theories outpace their experimental testing. I argue that this imbalance results from a combination of interdisciplinary gaps, a disparity in research approaches, and the limited accessibility of the theory to experimentalists. I call on interdisciplinary researchers to construct a road map for coexistence research, recommend the actions that should be implemented therein, and illustrate how multiple mechanisms can be integrated under the same coexistence terms, thus constructing a research landscape that can create uniformity in their experimental designs.     

In silico mapping of important genes and markers available in the public domain for efficient sorghum breeding

Crop genome sequencing projects generate massive amounts of genomic sequence information, and the utilization of this information in applied crop improvement programs has been augmented by the availability of sophisticated bioinformatics tools. Here, we present the possible direct utilization of sequence data from a sorghum genome sequencing project in applied crop breeding programs. Based on sequence homology, we aligned all publicly available simple sequence repeat markers on a sequence-based physical map for sorghum. Linking this physical map with already existing linkage map(s) provides better options for applied molecular breeding programs. When a new set of markers is made available, the new markers can be first aligned on a sequence-based physical map, and those located near the quantitative trait locus (QTL) can be identified from this map, thereby reducing the number of markers to be tested in order to identify polymorphic flanking markers for the QTL for any given donor × recurrent parent combination. Polymorphic markers that are expected (on the basis of their position on the sequence-based physical map) to be closely linked to the target can be used for foreground selection in marker-assisted breeding. This map facilitates the identification of a set of markers representing the entire genome, which would provide better resolution in diversity analyses and further linkage disequilibrium mapping. Filling the gaps in existing linkage maps and fine mapping can be achieved more efficiently by targeting the specific genomic regions of interest. It also opens up new exciting opportunities for comparative mapping and for the development of new genomic resources in related crops, both of which are lagging behind in the current genomic revolution. This paper also presents a number of examples of potential applications of sequence-based physical map for sorghum.     

Spatial-area selective retrieval of multiple object–place associations in a hierarchical cognitive map formed by theta phase coding

The human cognitive map is known to be hierarchically organized consisting of a set of perceptually clustered landmarks. Patient studies have demonstrated that these cognitive maps are maintained by the hippocampus, while the neural dynamics are still poorly understood. The authors have shown that the neural dynamic “theta phase precession” observed in the rodent hippocampus may be capable of forming hierarchical cognitive maps in humans. In the model, a visual input sequence consisting of object and scene features in the central and peripheral visual fields, respectively, results in the formation of a hierarchical cognitive map for object–place associations. Surprisingly, it is possible for such a complex memory structure to be formed in a few seconds. In this paper, we evaluate the memory retrieval of object–place associations in the hierarchical network formed by theta phase precession. The results show that multiple object–place associations can be retrieved with the initial cue of a scene input. Importantly, according to the wide-to-narrow unidirectional connections among scene units, the spatial area for object–place retrieval can be controlled by the spatial area of the initial cue input. These results indicate that the hierarchical cognitive maps have computational advantages on a spatial-area selective retrieval of multiple object–place associations. Theta phase precession dynamics is suggested as a fundamental neural mechanism of the human cognitive map.     

The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R ₀ is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.Dedicated to Karl Peter Hadeler on the occasion of his 70th birthday.     

The effect of an imprecise map on interval mapping QTLs

The statistical analysis of quantitative trait locus (QTL) experiments relies on the use of a linkage map of the markers genotyped. Such a map is, at best, a good estimate of the true map. Resources might be diverted into developing better marker maps or improved maps become available after the analysis, raising concerns over the original analysis. It is therefore important to understand the sensitivity of QTL analysis to map inaccuracy. We have used simulation methods to investigate the consequences of an incorrect map on the results of a QTL analysis using interval mapping. Backcross data sets were generated with a particular map and then analysed with both the correct map and incorrect maps. If the incorrect maps maintained the true linkage groups (i.e. no markers were incorrectly assigned to another linkage group), the accuracy of the map had little or no impact on the ability to detect QTLs, the true significance levels of the tests or the relative placement of QTLs. When a marker was incorrectly placed on another linkage group, there was a small increase in the level of the test. After adjusting for this increase, there was a decrease in power to detect a QTL near the misplaced marker. This decrease was of a similar magnitude to that found when using a single-marker analysis compared with interval mapping. These results mean that QTL analyses can proceed without the need for very accurate marker maps, and that estimated QTL positions can be translated onto updated maps without the need for reanalysis.     

Participatory Development and Analysis of a Fuzzy Cognitive Map of the Establishment of a Bio-Based Economy in the Humber Region

Fuzzy Cognitive Mapping (FCM) is a widely used participatory modelling methodology in which stakeholders collaboratively develop a ‘cognitive map’ (a weighted, directed graph), representing the perceived causal structure of their system. This can be directly transformed by a workshop facilitator into simple mathematical models to be interrogated by participants by the end of the session. Such simple models provide thinking tools which can be used for discussion and exploration of complex issues, as well as sense checking the implications of suggested causal links. They increase stakeholder motivation and understanding of whole systems approaches, but cannot be separated from an intersubjective participatory context. Standard FCM methodologies make simplifying assumptions, which may strongly influence results, presenting particular challenges and opportunities. We report on a participatory process, involving local companies and organisations, focussing on the development of a bio-based economy in the Humber region. The initial cognitive map generated consisted of factors considered key for the development of the regional bio-based economy and their directional, weighted, causal interconnections. A verification and scenario generation procedure, to check the structure of the map and suggest modifications, was carried out with a second session. Participants agreed on updates to the original map and described two alternate potential causal structures. In a novel analysis all map structures were tested using two standard methodologies usually used independently: linear and sigmoidal FCMs, demonstrating some significantly different results alongside some broad similarities. We suggest a development of FCM methodology involving a sensitivity analysis with different mappings and discuss the use of this technique in the context of our case study. Using the results and analysis of our process, we discuss the limitations and benefits of the FCM methodology in this case and in general. We conclude by proposing an extended FCM methodology, including multiple functional mappings within one participant-constructed graph.     

凉山半细毛羊1号染色体微卫星遗传连锁图谱的构建

实验选择绵羊1号条染色体上的9个微卫星标记,采用父系半同胞家系群体（共387个个体）构建凉山半细毛羊1号染色体遗传连锁图。建立的资源参考家系通过20个微卫星标记进行了系谱确证。试验结果表明,9个标记的等位基因数变化范围为5~15个,杂合度在0.202~0.831之间,平均杂合度为0.617,各标记的平均多态信息含量PIC=0.604。构建的凉山半细毛羊1号条染色体遗传连锁图总长度311.0 cM,与美国肉畜中心（USDA）和国际绵羊作图中心（IMF）构建的绵羊1号条染色遗传连锁图结果基本一致。可用于下一步的QTL定位研究。     

Alignment of molecular and classical linkage maps of rice,Oryza sativa

Application of genetic linkage maps in plant genetics and breeding can be greatly facilitated by integrating the available classical and molecular genetic linkage maps. In rice, Oryza sativa L., the classical linkage map includes about 300 genes which correspond to various important morphological, physiological, biochemical and agronomic characteristics. The molecular maps consist of more than 500 DNA markers which cover most of the genome within relatively short intervals. Little effort has been made to integrate these two genetic maps. In this paper we report preliminary results of an ongoing research project aimed at the complete integration and alignment of the two linkage maps of rice. Six different F₂ populations segregating for various phenotypic and RFLP markers were used and a total of 12 morphological and physiological markers (Table 1) were mapped onto our recently constructed molecular map. Six linkage groups (i.e., chr. 1, 3, 7, 9, 11 and 12) on our RFLP map were aligned with the corresponding linkage groups on the classical map, and the previous alignment for chromosome 6 was further confirmed by RFLP mapping of an additional physiological marker on this chromosome. Results from this study, combined with our previous results, indicate that, for most chromosomes in rice, the RFLP map encompasses the classical map. The usefulness of an integrated genetic linkage map for rice genetics and breeding is discussed.Abbreviations RFLP restriction fragment length polymorphism - chr chromosome - cM centiMorgan     

Probing the Flexibility of Large Conformational Changes in Protein Structures through Local Perturbations

Protein conformational changes and dynamic behavior are fundamental for such processes as catalysis, regulation, and substrate recognition. Although protein dynamics have been successfully explored in computer simulation, there is an intermediate-scale of motions that has proven difficult to simulate—the motion of individual segments or domains that move independently of the body the protein. Here, we introduce a molecular-dynamics perturbation method, the Rotamerically Induced Perturbation (RIP), which can generate large, coherent motions of structural elements in picoseconds by applying large torsional perturbations to individual sidechains. Despite the large-scale motions, secondary structure elements remain intact without the need for applying backbone positional restraints. Owing to its computational efficiency, RIP can be applied to every residue in a protein, producing a global map of deformability. This map is remarkably sparse, with the dominant sites of deformation generally found on the protein surface. The global map can be used to identify loops and helices that are less tightly bound to the protein and thus are likely sites of dynamic modulation that may have important functional consequences. Additionally, they identify individual residues that have the potential to drive large-scale coherent conformational change. Applying RIP to two well-studied proteins, Dihdydrofolate Reductase and Triosephosphate Isomerase, which possess functionally-relevant mobile loops that fluctuate on the microsecond/millisecond timescale, the RIP deformation map identifies and recapitulates the flexibility of these elements. In contrast, the RIP deformation map of α-lytic protease, a kinetically stable protein, results in a map with no significant deformations. In the N-terminal domain of HSP90, the RIP deformation map clearly identifies the ligand-binding lid as a highly flexible region capable of large conformational changes. In the Estrogen Receptor ligand-binding domain, the RIP deformation map is quite sparse except for one large conformational change involving Helix-12, which is the structural element that allosterically links ligand binding to receptor activation. RIP analysis has the potential to discover sites of functional conformational changes and the linchpin residues critical in determining these conformational states.     

Linkage of a Known Chloroplast Gene Mutation to the Uniparental Genome of CHLAMYDOMONAS REINHARDII

Data are presented that associate three new markers with the uniparental linkage group in Chlamydomonas reinhardii. One of these, mutant 10-6C, is a genetic marker for the structural gene of the large subunit of ribulose bisphosphate carboxylase. These results provide the first direct link between the uniparental gene map and the physical map of chloroplast DNA. The other two markers, Dr2 (DCMU resistant) and 8-36C (deficient in photosystem II activity), map to a single locus. The data suggest that mixing in zygotic chloroplasts may not be complete so that input genomes do not have equal opportunities to recombine. The data are not compatible with simple linear or circular maps but can be explained on the basis of the known physical structure of chloroplast DNA.     

Toward the mapping of physiological and agronomic characters on a rice function map: QTL analysis and comparison between QTLs and expressed sequence tags

We have constructed a rice function map by collating the results on quantitative trait loci (QTLs) for 23 important physiological and agronomic characters (including 13 newly measured traits) obtained using backcross inbred lines of japonica Nipponbare×indica Kasalath. Using these materials, The Rice Genome project (RGP) has developed a high-density genetic map. QTLs controlling yield did not overlap with those controlling the morphological and physiological traits supposed to relate to yield, such as photosynthetic ability. This result suggests that these traits do not influence yield, at least in this genetic background and environment. QTLs controlling yield also did not overlap with the structural genes controlling carbon metabolism (rbcS, cytosolic or plastidic fructose-1,6-bisphosphate, R-enzyme, and sucrose synthase).The combination of a function map and results from the RGP can be advantageous. The utility of this map is discussed. Received: 1 October 1999 / Accepted: 28 July 2000