The relationship between real-time and discrete-generation models of epidemic spread

Many important results in stochastic epidemic modelling are based on the Reed-Frost model or on other similar models that are characterised by unrealistic temporal dynamics. Nevertheless, they can be extended to many other more realistic models thanks to an argument first provided by Ludwig [Final size distributions for epidemics, Math. Biosci. 23 (1975) 33-46], that states that, for a disease leading to permanent immunity after recovery, under suitable conditions, a continuous-time infectious process has the same final size distribution as another more tractable discrete-generation contact process; in other words, the temporal dynamics of the epidemic can be neglected without affecting the final size distribution. Despite the importance of such an argument, its presence behind many results is often not clearly stated or hidden in references to previous results. In this paper, we reanalyse Ludwig’s result, highlighting some of the conditions under which it does not hold and providing a general framework to examine the differences between the continuous-time and the discrete-generation process.     

The relation between synapsis and chiasma formation in Rhoeo spathacea

Rhoeo spathacea (Schwartz) Stearn (2n=12) is a complete translocation heterozygote in which a ring of 12 chromosomes is often observed from diakinesis through metaphase I. A new hypotonie bursting technique on primary microsporocytes made possible visualization of the complete pattern of pachytene synapsis in R. spathacea by light and electron microscopy. Analysis showed that most of the length of a chromosome never synapsed and that relational twisting of the lateral elements was prominent in the distal segments of most synaptonemal complexes (SCs). The most common situation was 12 relatively short SCs representing the 24 synapsed terminal segments of the 12 chromosomes; 11 and 10 SCs were also observed but less frequently, indicating one and two synaptic failures, respectively. Similarly, in diakinesis a ring of 12 chromosomes was most common, but a chain of 12 chromosomes and two chains of chromosomes could also occur. These situations indicated zero, one, and two chiasma failures, respectively. The frequencies of synaptic failure (0.021) and of chiasma failure (0.022) were essentially the same as were the distributions of cells with different numbers of synaptic and chiasma failures. These comparisons indicate that if synapsis occurs between two distal segments of the chromosomes of R. spathacea, a chiasma will almost certainly follow. A diagram to illustrate the arrangement of mid-pachytene chromosomes and a model to explain the origin of relational twisting of lateral elements in SCs are presented.     

Relationship between incomplete synapsis and chiasma localization

One of the subjects within the meiotic field that has been actively investigated in the recent years is the temporal and functional relationships between meiotic recombination, cohesin loading and synaptonemal complex (SC) assembly. Although the study of meiotic mutants has shed some light, many questions remain to be answered. Here, we have studied this topic in the orthopteran Paratettix meridionalis, a species with telocentric chromosomes, which shows two unusual cytological features: pairing and synapsis of homologues during prophase I are restricted to the non-centromeric distal regions and extremely distal chiasma localization in metaphase I bivalents. In order to determine whether there is a relationship between both phenomena, we have used: (1) a spreading technique for following the ultrastructure of SC assembly and (2) immunofluorescence for SMC3 and SMC1α cohesin subunits, which mark the development of the axial element (a SC component); the histone γ-H2AX, which mostly labels the sites of double-strand breaks; and the recombinase RAD51. Spermatocytes showed conspicuous polarization of both the maturation of cohesin axes and the initiation of meiotic recombination events. Consequently, it is proposed that maturation of cohesin axes, which begins in very distal regions, could drive the latter loading of recombinases to such regions. This restricted distribution of recombination events along homologues would finally be responsible for the incomplete pairing and synapsis observed in all autosomes of the complement and hence for chiasma localization. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. All authors contributed equally to this work.     

Discrete chiasma formation models and their associated high order interference

. We introduce some special chiasma formation processes. First a family of discrete chiasma formation processes is introduced and we determine the nature of higher order interference associated with those processes. Secondly we consider a two-stage chiasma formation process, where the associated recombination frequency between two markers depends not only on their map distance but also on their location along the chromosomes. We characterise under this process, in some cases, the nature of interference between two segments. Received: 22 January 1996 / Revised version: 17 September 1997     

On the relationship between two models of neural entrainment

A comparison is established between the two models of neural entrainment proposed by Nagumo and Sato (1972) and by Torras (1986). Both are formulated in terms of a transition equation whose solutions of period m correspond to the different (m:r)-entrainment patterns. The existence of a transformation relating both transition equations implies the equivalence between stable entrainment for the former model and unstable entrainment for the latter one. However, there is no counterpart for the stable entrainment arising in Torras' model.     

Simple models for complex systems: exploiting the relationship between local and global densities

Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.     

Modal analysis of renewal models for spontaneous single neuron discharge

Three renewal process models of neural discharge were briefly discussed, and their modal properties were analyzed. Examples of numerical solutions for the p · d · f's of each model were presented, and they conformed with the analytical results, demonstrating that one of the models, Model 2, generates multimodal interresponse time p · d · f's. Several areas of comparison with real data were indicated.Supported in part by a grant from the Alfred P. Sloan Foundation to the Department of Theoretical Biology, and by Research Grant No. NSF-GP-16071 from the Division of Mathematical, Physical and Engineering Sciences of the National Science Foundation to the Department of Statistics, University of Chicago.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.     

The relation between pairing preference and chiasma frequency in tetrasomics of rye.

The association pattern of marked tetrasomes of Secale chromosome 1R at meiotic first metaphase was analyzed. Two of the four chromosomes were identical with terminal C-bands at both arms; the other two were also identical but lacked C-bands and were homologous or homeologous with the first two. Four different types of heterozygotes for 1R were studied: (i). autotetraploid hybrids between genetic variants within Secale cereale subsp. cereale, (ii). tetraploid hybrids between subspecies of Secale cereale, (iii). tetraploid hybrids between species of Secale, and (iv). autotetrasomes of S. cereale in a wheat background. Earlier observations that heterozygous associations (banded with unbanded) had consistently higher chiasma frequencies than homozygous associations were extended and confirmed. To analyze this phenomenon more closely, the possible relations between this correlation and several other meiotic phenomena were studied. For this analysis, three genetically different autotetraploid hybrids within S. cereale were selected that differed with respect to the relation between pairing type and chiasma frequency. Special attention was given to different patterns of interference and other meiotic phenomena in the two chromosome arms of chromosome 1R. No relations between such phenomena and the relation between pairing type and chiasma frequency could be established. A hypothesis is formulated assuming that long-distance homologue attraction is concentrated in a limited number of sites and that in different genotypes, different patterns of active sites are present. Moderately weak attraction sites can pair with strong homologous sites under favorable genetic conditions, but two weak sites cannot. Then, heterozygotes have more effective pairing initiation and consequently chiasma formation than homozygotes. Under less favorable conditions, only strong sites are effective, and then, homozygotes pair better, but the chiasma frequency is lower. A model of the forces involved in homologue attraction is presented.     

The misuse of the three-point cross for estimating chiasma interference

The building of genetic maps in diploid organisms by crosses between different genotypes and estimation of recombination frequencies from the obtained segregation data has been successfully used since a very young step in the birth of genetics. The three-point cross methodology has facilitated this task and has demonstrated at the same time that genetic distances are not additive, as some recombinant products are not recognised in the progeny. Three-point cross also allows to examine if chiasma interference exists and its evaluation. Here I show that the classical method of this estimation is erroneous and inevitably determines the apparition of a spurious, positive interference, which has been claimed to be an almost general phenomenon. Interference can only be estimated from a precise knowledge of the number of crossing over events occurring in meiocytes.     

The relationship between structure and function for the sulfite reductases

The six-electron reductions of sulfite to sulfide and nitrite to ammonia, fundamental to early and contemporary life, are catalyzed by diverse sulfite and nitrite reductases that share an unusual prosthetic assembly in their active centers, namely siroheme covalently linked to an Fe₄S₄ cluster. The recently determined crystallographic structure of the sulfite reductase hemoprotein from Escherichia coli complements extensive biochemical and spectroscopic studies in revealing structural features that are key for the catalytic mechanism and in suggesting a common symmetric structural unit for this diverse family of enzymes.     

乙型肝炎血清标志模式与病毒载量的关系及意义

为探讨乙型肝炎 (以下简称乙肝 )血清标志模式与病毒载量的关系及临床意义 ,作者选择符合 2 0 0 0年《全国病毒性肝炎诊断标准》的慢性肝炎血清 1343份 ,分别用ELISA法、PCR ELISA法检测HB血清标志、HBV -DNA和 1896位点变异株。结果显示 :HBsAg阳性血清 10 97份 (81.6 8%、)HBsAg阴性血清 2 4 6份 (81.31% )。在HBsAg阳性血清中 ,HBsAg、HBeAg、抗 HBc(1 3 5 )阳性组 4 0 4份 (30 .l% ) ,HBV -DNA阳性 347份 (85 .89% ) ,DNA阳性值呈递增趋势 (10 4～ 10 6拷贝 /ml,各占 8.6 5 %、33.71%、5 3.6 1% ) ;而HBsAg、抗 HBe阳性组血清 6 0 6份 (45 .12 % ) ,DNA阳性值 10 5拷贝 /ml,占优势 (6 4 .18% )。在HBsAg阴性血清中 ,抗 HBs、抗 HBe、抗 HBc(2 4 5 )阳性组 2 32份 (17.2 7% ) ,DNA阳性占 7.32 % ,DNA阳性值递减由 10 4～ 10 6拷贝 /ml,各占 5 2 .9%、4 1.l%、5 .9%。结论 ,各血清标志模式中的病毒载量为HBsAg阳性组 >HBsAg阴性组 ,阳性组 1 3 5 >1 4 5 >l 5 >2 4 5。但 1 3 5阳性组中 14 %在界值以下 ,1 4 5阳性组中 1896位点自然变异达 78.6 % ,2 4 5阳性组中仍存在DNA+ 血清 ,以上提示在临床判定和治疗时要慎重对待(注 :l=HBsAg,2 =抗 HBs,3=HBeAg ,4 =抗 HBe ,5 =抗 HBc)     

Enhanced process understanding and multivariate prediction of the relationship between cell culture process and monoclonal antibody quality

This work investigates the insights and understanding which can be deduced from predictive process models for the product quality of a monoclonal antibody based on designed high‐throughput cell culture experiments performed at milliliter (ambr‐15^®) scale. The investigated process conditions include various media supplements as well as pH and temperature shifts applied during the process. First, principal component analysis (PCA) is used to show the strong correlation characteristics among the product quality attributes including aggregates, fragments, charge variants, and glycans. Then, partial least square regression (PLS1 and PLS2) is applied to predict the product quality variables based on process information (one by one or simultaneously). The comparison of those two modeling techniques shows that a single (PLS2) model is capable of revealing the interrelationship of the process characteristics to the large set product quality variables. In order to show the dynamic evolution of the process predictability separate models are defined at different time points showing that several product quality attributes are mainly driven by the media composition and, hence, can be decently predicted from early on in the process, while others are strongly affected by process parameter changes during the process. Finally, by coupling the PLS2 models with a genetic algorithm first the model performance can be further improved and, most importantly, the interpretation of the large‐dimensioned process–product‐interrelationship can be significantly simplified. The generally applicable toolset presented in this case study provides a solid basis for decision making and process optimization throughout process development. © 2017 American Institute of Chemical Engineers Biotechnol. Prog., 33:1368–1380, 2017