Accuracy of rate estimation using relaxed-clock models with a critical focus on the early metazoan radiation

In recent years, a number of phylogenetic methods have been developed for estimating molecular rates and divergence dates under models that relax the molecular clock constraint by allowing rate change throughout the tree. These methods are being used with increasing frequency, but there have been few studies into their accuracy. We tested the accuracy of several relaxed-clock methods (penalized likelihood and Bayesian inference using various models of rate change) using nucleotide sequences simulated on a nine-taxon tree. When the sequences evolved with a constant rate, the methods were able to infer rates accurately, but estimates were more precise when a molecular clock was assumed. When the sequences evolved under a model of autocorrelated rate change, rates were accurately estimated using penalized likelihood and by Bayesian inference using lognormal and exponential models of rate change, while other models did not perform as well. When the sequences evolved under a model of uncorrelated rate change, only Bayesian inference using an exponential rate model performed well. Collectively, the results provide a strong recommendation for using the exponential model of rate change if a conservative approach to divergence time estimation is required. A case study is presented in which we use a simulation-based approach to examine the hypothesis of elevated rates in the Cambrian period, and it is found that these high rate estimates might be an artifact of the rate estimation method. If this bias is present, then the ages of metazoan divergences would be systematically underestimated. The results of this study have implications for studies of molecular rates and divergence dates.

Time dependency of molecular rate estimates and systematic overestimation of recent divergence times

Studies of molecular evolutionary rates have yielded a wide range of rate estimates for various genes and taxa. Recent studies based on population-level and pedigree data have produced remarkably high estimates of mutation rate, which strongly contrast with substitution rates inferred in phylogenetic (species-level) studies. Using Bayesian analysis with a relaxed-clock model, we estimated rates for three groups of mitochondrial data: avian protein-coding genes, primate protein-coding genes, and primate d-loop sequences. In all three cases, we found a measurable transition between the high, short-term (<1–2 Myr) mutation rate and the low, long-term substitution rate. The relationship between the age of the calibration and the rate of change can be described by a vertically translated exponential decay curve, which may be used for correcting molecular date estimates. The phylogenetic substitution rates in mitochondria are approximately 0.5% per million years for avian protein-coding sequences and 1.5% per million years for primate protein-coding and d-loop sequences. Further analyses showed that purifying selection offers the most convincing explanation for the observed relationship between the estimated rate and the depth of the calibration. We rule out the possibility that it is a spurious result arising from sequence errors, and find it unlikely that the apparent decline in rates over time is caused by mutational saturation. Using a rate curve estimated from the d-loop data, several dates for last common ancestors were calculated: modern humans and Neandertals (354 ka; 222–705 ka), Neandertals (108 ka; 70–156 ka), and modern humans (76 ka; 47–110 ka). If the rate curve for a particular taxonomic group can be accurately estimated, it can be a useful tool for correcting divergence date estimates by taking the rate decay into account. Our results show that it is invalid to extrapolate molecular rates of change across different evolutionary timescales, which has important consequences for studies of populations, domestication, conservation genetics, and human evolution.

Evidence for time dependency of molecular rate estimates

Long-term changes in the genetic composition of a population occur by the fixation of new mutations, a process known as substitution. The rate at which mutations arise in a population and the rate at which they are fixed are expected to be equal under neutral conditions (Kimura, 1968). Between the appearance of a new mutation and its eventual fate of fixation or loss, there will be a period in which it exists as a transient polymorphism in the population (Kimura and Ohta, 1971). If the majority of mutations are deleterious (and nonlethal), the fixation probabilities of these transient polymorphisms are reduced and the mutation rate will exceed the substitution rate (Kimura, 1983). Consequently, different apparent rates may be observed on different time scales of the molecular evolutionary process (Penny, 2005; Penny and Holmes, 2001). The substitution rate of the mitochondrial protein-coding genes of birds and mammals has been traditionally recognized to be about 0.01 substitutions/site/million years (Myr) (Brown et al., 1979; Ho, 2007; Irwin et al., 1991; Shields and Wilson, 1987), with the noncoding D-loop evolving several times more quickly (e.g., Pesole et al., 1992; Quinn, 1992). Over the past decade, there has been mounting evidence that instantaneous mutation rates substantially exceed substitution rates, in a range of organisms (e.g., Denver et al., 2000; Howell et al., 2003; Lambert et al., 2002; Mao et al., 2006; Mumm et al., 1997; Parsons et al., 1997; Santos et al., 2005). The immediate reaction to the first of these findings was that the polymorphisms generated by the elevated mutation rate are short-lived, perhaps extending back only a few hundred years (Gibbons, 1998; Macaulay et al., 1997). That is, purifying selection was thought to remove these polymorphisms very rapidly.

Use of Bayesian MUNE to show differing rate of loss of motor units in subgroups of ALS

Objectives To evaluate differences among patients with different clinical features of ALS, we used our Bayesian method of motor unit number estimation (MUNE). Methods We performed serial MUNE studies on 42 subjects who fulfilled the diagnostic criteria for ALS during the course of their illness. Subjects were classified into three subgroups according to whether they had typical ALS (with upper and lower motor neurone signs) or had predominantly upper motor neurone weakness with only minor LMN signs, or predominantly lower motor neurone weakness with only minor UMN signs. In all subjects we calculated the half life of MUs, defined as the expected time for the number of MUs to halve, in one or more of the abductor digiti minimi (ADM), abductor pollicis brevis (APB) and extensor digitorum brevis (EDB) muscles. Results The mean half life of MUs was less in subjects who had typical ALS with both upper and lower motor neurone signs than in those with predominantly upper motor neurone weakness or predominantly lower motor neurone weakness. In 18 subjects we analysed the estimated size of the MUs and demonstrated the appearance of large MUs in subjects with upper or lower motor neurone predominant weakness. We found that the appearance of large MUs was correlated with the half life of MUs. Conclusions Patients with different clinical features of ALS have different rates of loss and different sizes of MUs. Significance: These findings could indicate differences in disease pathogenesis.

Environmental factors controlling temporal and spatial variability in the soil-atmosphere exchange of CO2, CH4 and N2O from an Australian subtropical rainforest

The temporal variations in CO2, CH4 and N2O fluxes were measured over two consecutive years from February 2007 to March 2009 from a subtropical rainforest in south-eastern Queensland, Australia, using an automated sampling system. A concurrent study using an additional 30 manual chambers examined the spatial variability of emissions distributed across three nearby remnant rainforest sites with similar vegetation and climatic conditions. Interannual variation in fluxes of all gases over the 2 years was minimal, despite large discrepancies in rainfall, whereas a pronounced seasonal variation could only be observed for CO2 fluxes. High infiltration, drainage and subsequent high soil aeration under the rainforest limited N2O loss while promoting substantial CH4 uptake. The average annual N2O loss of 0.5 ± 0.1 kg N2O-N ha−1 over the 2-year measurement period was at the lower end of reported fluxes from rainforest soils. The rainforest soil functioned as a sink for atmospheric CH4 throughout the entire 2-year period, despite periods of substantial rainfall. A clear linear correlation between soil moisture and CH4 uptake was found. Rates of uptake ranged from greater than 15 g CH4-C ha−1 day−1 during extended dry periods to less than 2–5 g CH4-C ha−1 day−1 when soil water content was high. The calculated annual CH4 uptake at the site was 3.65 kg CH4-C ha−1 yr−1. This is amongst the highest reported for rainforest systems, reiterating the ability of aerated subtropical rainforests to act as substantial sinks of CH4. The spatial study showed N2O fluxes almost eight times higher, and CH4 uptake reduced by over one-third, as clay content of the rainforest soil increased from 12% to more than 23%. This demonstrates that for some rainforest ecosystems, soil texture and related water infiltration and drainage capacity constraints may play a more important role in controlling fluxes than either vegetation or seasonal variability

Relative entropy rate based model selection for linear hybrid system filters of uncertain nonlinear systems

Hybrid system representations have been exploited in a number of challenging modelling situations, including situations where the original nonlinear dynamics are too complex (or too imprecisely known) to be directly filtered. Unfortunately, the question of how to best design suitable hybrid system models has not yet been fully addressed, particularly in the situations involving model uncertainty. This paper proposes a novel joint state-measurement relative entropy rate based approach for design of hybrid system filters in the presence of (parameterised) model uncertainty. We also present a design approach suitable for suboptimal hybrid system filters. The benefits of our proposed approaches are illustrated through design examples and simulation studies.

Quantum key distribution in the classical authenticated key exchange framework

Key establishment is a crucial primitive for building secure channels in a multi-party setting. Without quantum mechanics, key establishment can only be done under the assumption that some computational problem is hard. Since digital communication can be easily eavesdropped and recorded, it is important to consider the secrecy of information anticipating future algorithmic and computational discoveries which could break the secrecy of past keys, violating the secrecy of the confidential channel. Quantum key distribution (QKD) can be used generate secret keys that are secure against any future algorithmic or computational improvements. QKD protocols still require authentication of classical communication, although existing security proofs of QKD typically assume idealized authentication. It is generally considered folklore that QKD when used with computationally secure authentication is still secure against an unbounded adversary, provided the adversary did not break the authentication during the run of the protocol. We describe a security model for quantum key distribution extending classical authenticated key exchange (AKE) security models. Using our model, we characterize the long-term security of the BB84 QKD protocol with computationally secure authentication against an eventually unbounded adversary. By basing our model on traditional AKE models, we can more readily compare the relative merits of various forms of QKD and existing classical AKE protocols. This comparison illustrates in which types of adversarial environments different quantum and classical key agreement protocols can be secure.