Derivation of a Floquet formalism within a natural framework.

Many biological systems experience a periodic environment. Floquet theory is a mathematical tool to deal with such time periodic systems. It is not often applied in biology, because linkage between the mathematics and the biology is not available. To create this linkage, we derive the Floquet theory for natural systems. We construct a framework, where the rotation of the Earth is causing the periodicity. Within this framework the angular momentum operator is introduced to describe the Earth's rotation. The Fourier operators and the Fourier states are defined to link the rotation to the biological system. Using these operators, the biological system can be transformed into a rotating frame in which the environment becomes static. In this rotating frame the Floquet solution can be derived. Two examples demonstrate how to apply this natural framework.

Risk of poultry compartments for transmission of Highly Pathogenic Avian Influenza.

When outbreaks of Highly Pathogenic Avian Influenza (HPAI) occur in OIE member countries with until then disease-free status, member countries can use 'compartmentalisation'. A compartment may be defined as a subset of farms under a common management system, complying with certain stringent surveillance, control and biosecurity measures, and based on that may receive a disease-free status. Based on this disease-free status the compartment is exempted from certain transport restrictions coming into force in case of outbreaks occurring in the country. For deciding whether a candidate compartment is granted official compartment status, it is relevant to assess the additional HPAI transmission risks that would arise due to the exemptions granted. These risks consist of both additional local transmission risks as well as the additional risk of a 'jump' of HPAI infection from one poultry area, via the compartment, to another area. Here such risk assessment is carried out using a spatial mathematical model for between-farm transmission in the Netherlands, yielding insight in the roles of compartment composition and contact structure and identify relevant evaluation criteria for granting HPAI compartment status. At the core of this model are transmission probabilities associated with indirect between-farm contacts, e.g. through feed delivery, egg collection and professional visitors. These probabilities were estimated from Dutch epidemic outbreak data in earlier work. The additional risk of a jump of HPAI from one area, via the compartment, to another area is calculated relative to the direct jump risk. The results show that additional transmission risks caused by a compartment to other farms are mainly dependent on the distance of densely populated poultry areas (DPPAs) to the nearest compartment farm. Apart from conditions on these distances, we also recommend specific routing requirements for transport and other movements within the compartment.

Estimating the between-farm transmission rates for highly pathogenic avian influenza subtype H5N1 epidemics in Bangladesh between 2007 and 2013.

Highly Pathogenic Avian Influenza (HPAI) is classified by the World Organization for Animal Health as one of the notifiable diseases. Its occurrence is associated with severe socio-economic impacts and is also zoonotic. Bangladesh HPAI epidemic data for the period between 2007 and 2013 were obtained and split into epidemic waves based on the time lag between outbreaks. By assuming the number of newly infected farms to be binomially distributed, we fit a Generalized Linear Model to the data to estimate between-farm transmission rates (ß). These parameters are then used together with the calculated infectious periods to estimate the respective basic reproduction numbers (R0 ). The change in ß and R0 with time during the course of each epidemic wave was explored. Finally, sensitivity analyses of the effects of reducing the delay in detecting infection on a farm as well as extended infectiousness of a farm beyond the day of culling were assessed. The point estimates obtained for ß ranged from 0.08 (95% CI: 0.06-0.10) to 0.11 (95% CI: 0.08-0.20) per infectious farm per day while R0 ranged from 0.85 (95% CI: 0.77-1.02) to 0.96 (95% CI: 0.72-1.20). Sensitivity analyses reveal that the estimates are quite robust to changes in the assumptions about the day in reporting infection and extended infectiousness. In the analysis allowing for time-varying transmission parameters, the rising and declining phases observed in the epidemic data were synchronized with the moments when R0 was greater and less than one, respectively. From an epidemiological perspective, the consistency of these estimates and their magnitude (R0  ≈ 1) indicate that the effectiveness of the deployed control measures was largely invariant between epidemic waves and the trend of the time-varying R0 supports the hypothesis of sustained farm-to-farm transmission that is possibly initiated by a few unique introductions.

Proton MAS NMR spectra at high magnetic fields and high spinning frequencies: spectral simulations using floquet theory.

Proton magic angle spinning (MAS) spectra of a model spin system, consisting of six protons, were calculated for different values of the external magnetic field and the spinning frequencies. Floquet theory was used to evaluate these spectra. The reduction of the effective homonuclear dipolar interaction for increasing spinning frequency was investigated. The influence of an increase of the external magnetic field and the spinning frequencies on the linewidths of the centerband spectra is discussed. This Floquet description of the rotating proton spin system will assist us in our calculations of the CPMAS spin dynamics of a low abundant spin interacting with a set of coupled protons.

Risk based surveillance for early detection of low pathogenic avian influenza outbreaks in layer chickens.

Current knowledge does not allow the prediction of when low pathogenic avian influenza virus (LPAIV) of the H5 and H7 subtypes infecting poultry will mutate to their highly pathogenic phenotype (HPAIV). This mutation may already take place in the first infected flock; hence early detection of LPAIV outbreaks will reduce the likelihood of pathogenicity mutations and large epidemics. The objective of this study was the development of a model for the design and evaluation of serological-surveillance programmes, with a particular focus on early detection of LPAIV infections in layer chicken flocks. Early detection is defined as the detection of an infected flock before it infects on average more than one other flock (between-flock reproduction ratio Rf<1), hence a LPAI introduction will be detected when only one or a few other flocks are infected. We used a mathematical model that investigates the required sample size and sampling frequency for early detection by taking into account the LPAIV within- and between-flock infection dynamics as well as the diagnostic performance of the serological test used. Since layer flocks are the target of the surveillance, we also explored whether the use of eggs, is a good alternative to sera, as sample commodity. The model was used to refine the current Dutch serological-surveillance programme. LPAIV transmission-risk maps were constructed and used to target a risk-based surveillance strategy. In conclusion, we present a model that can be used to explore different sampling strategies, which combined with a cost-benefit analysis would enhance surveillance programmes for low pathogenic avian influenza.

Mapping the basic reproduction number (R0) for vector-borne diseases: a case study on bluetongue virus.

Geographical maps indicating the value of the basic reproduction number, R0, can be used to identify areas of higher risk for an outbreak after an introduction. We develop a methodology to create R0 maps for vector-borne diseases, using bluetongue virus as a case study. This method provides a tool for gauging the extent of environmental effects on disease emergence. The method involves integrating vector-abundance data with statistical approaches to predict abundance from satellite imagery and with the biologically mechanistic modelling that underlies R0. We illustrate the method with three applications for bluetongue virus in the Netherlands: 1) a simple R0 map for the situation in September 2006, 2) species-specific R0 maps based on satellite-data derived predictions, and 3) monthly R0 maps throughout the year. These applications ought to be considered as a proof-of-principle and illustrations of the methods described, rather than as ready-to-use risk maps. Altogether, this is a first step towards an integrative method to predict risk of establishment of diseases based on mathematical modelling combined with a geographic information system that may comprise climatic variables, landscape features, land use, and other relevant factors determining the risk of establishment for bluetongue as well as of other emerging vector-borne diseases.

CP-MAS 13C-NMR dipolar correlation spectroscopy of 13C-enriched chlorosomes and isolated bacteriochlorophyll c aggregates of Chlorobium tepidum: the self-organization of pigments is the main structural feature of chlorosomes.

Magic angle spinning (MAS) NMR dipolar correlation spectroscopy was applied for the first time to a biologically intact system, the light-harvesting chlorosomes of the green photosynthetic bacterium Chlorobium tepidum. The MAS spectra provide evidence that the self-organization of many thousands of bacteriochlorophyll c (BChl c) molecules is the predominant structural feature of the chlorosome. 13C-Enriched chlorosomes were prepared from nonuniformly labeled cultures grown with NaH13CO3 as the main carbon source and from a uniformly 13C-labeled culture grown with NaH13CO3 as the sole carbon source. For the nonuniformly labeled samples, the positions of the chlorin macrocycle originating from C-4 and C-5 of 5-aminolevulinic acid contained > 95% 13C while the remaining positions, which could have originated also from unlabeled acetate, were labeled to approximately 60% with 13C. The 1-D and 2-D MAS data of the labeled chlorosomes, when compared with data on the isolated labeled BChl c aggregated in n-hexane, show that the major component of the MAS signals in the chlorosomes is from BChl c, and only minor signal contributions arise from lipids and proteins. The 13C MAS signals of the BChl c aggregates were fully assigned by MAS 2-D dipolar correlation spectroscopy, using data on monomeric BChl c in CDCl3/CD3OD as reference. The 2(1)-, 3-, 3(2-), 5-, 12(1)-, 13-, and 13(1)-carbons are shifted by 2.5 ppm or more upfield with respect to the solution data.(ABSTRACT TRUNCATED AT 250 WORDS)

13C magic angle spinning NMR characterization of the functionally asymmetric QA binding in Rhodobacter sphaeroides R26 photosynthetic reaction centers using site-specific 13C-labeled ubiquinone-10.

Photosynthetic reaction centers (RCs) of Rhodobacter sphaeroides R26 were reconstituted at the QA site with ubiquinone-10, selectively 13C-enriched on positions 1, 2, 3, 4, and 3-Me (IUPAC numbering). RCs dispersed in LDAO detergent were studied with 13C CP/MAS NMR spectroscopy at temperatures between 180 and 240 K, while RCs precipitated by removal of the detergent were investigated at ambient temperature and at temperatures down to 180 K. Electrostatic charge differences in QA induced by polarization from the protein are less than 0.02 electronic equivalent for any of the labeled positions. This includes the 4-carbonyl, which is therefore not significantly polarized by an electrostatic binding interaction with the protein. The QA site is slightly heterogeneous on the scale of the NMR as the observed line widths of the labels are between 150 and 300 Hz and inhomogeneous broadening is observed for the signals of positions 1, 2, and 3 upon cooling. This contrasts with earlier MAS observations for labels in the vicinity of the special pair. The chemical shifts are 184, 144, and 137 ppm for the labels at positions 1, 2, 3, and 12 ppm for the 3-methyl 13C. For the 4-carbonyl only at sample temperatures below approximately 255 K a CP/MAS response can be observed at 183 ppm. The principal components of the chemical shift tensors for the ring labels in QA were estimated using difference spectroscopy.(ABSTRACT TRUNCATED AT 250 WORDS)

A refined model of the chlorosomal antennae of the green bacterium Chlorobium tepidum from proton chemical shift constraints obtained with high-field 2-D and 3-D MAS NMR dipolar correlation spectroscopy.

Heteronuclear 2-D and 3-D magic-angle spinning NMR dipolar correlation spectroscopy was applied to determine solid-state (1)H shifts for aggregated bacteriochlorophyll c (BChl c) in uniformly (13)C-enriched light harvesting chlorosomes of the green photosynthetic bacterium Chlorobium tepidum. A complete assignment of 29 different observable resonances of the 61 protons of the aggregated BChl c in the intact chlorosomes is obtained. Aggregation shifts relative to monomeric BChl c in solution are detected for protons attached to rings I, II, and III/V and to their side chains. The 2(1)-H(3), 3(2)-H(3), and 3(1)-H resonances are shifted upfield by -2.2, -1, and -3.3 ppm, respectively, relative to monomeric BChl c in solution. Although the resonances are inhomogeneously broadened and reveal considerable global structural heterogeneity, the 5-CH and the 7-Me responses are doubled, which provides evidence for the existence of at least two relatively well-defined structurally different arrangements. Ab initio quantum chemical modeling studies were performed to refine a model for the self-assembled BChl c with two different types of BChl stacks. The BChl in the stacks can adopt either anti- or syn-configuration of the coordinative bond, where anti and syn designate the relative orientation of the Mg-OH bond relative to the direction of the 17-17(1) bond. The analogy between aggregation shifts for BChl c in the chlorosome and for self-assembled chlorophyll a/H(2)O is explored, and a bilayer model for the tubular supra-structure of sheets of BChl c is proposed, from a homology modeling approach.

Characterization of pheophytin ground states in Rhodobacter sphaeroides R26 photosynthetic reaction centers from multispin pheophytin enrichment and 2-D 13C MAS NMR dipolar correlation spectroscopy.

The electronic ground states of pheophytin cofactors potentially involved in symmetry breaking between the A and B branch for electron transport in the bacterial photosynthetic reaction center have been investigated through a characterization of the electron densities at individual atomic positions of pheophytin a from 13C chemical shift data. A new experimental approach involving multispin 13C labeling and 2-D NMR is presented. Bacterial photosynthetic reaction centers of Rhodobacter sphaeroides R26 were reconstituted with uniformly 13C biosynthetically labeled (plant) Pheo a in the two pheophytin binding sites. From the multispin labeled samples 1-D and 2-D solid-state 13C magic angle spinning NMR spectra could be obtained and used to characterize the pheophytin a ground state in the Rb. sphaeroides R26 RCs, i.e., without a necessity for time-consuming selective labeling strategies involving organic synthesis. From the 2-D solid state 13C-13C correlation spectra collected with spinning speeds of 8 and 10 kHz, with mixing times of 1 and 0.8 ms, many 13C resonances of the [U-13C]Pheo a molecules reconstituted in the RCs could be assigned in a single set of experiments. Parts of the pheophytins interacting with the protein, at the level of 13C shifts modified by binding, could be identified. Small reconstitution shifts are detected for the 17(2) side chain of ring IV. In contrast, there is no evidence for electrostatic differences between the two Pheo a, for instance, due to a possibly strong selective electrostatic interaction with Glu L104 on the active branch. The protonation states appear the same, and the NMR suggests a strong overall similarity between the ground states of the two Pheo a, which is of interest in view of the asymmetry of the electron transfer.