Enhanced diffusion and the eigenvalue band structure of Brownian motion in tilted periodic potentials.

We consider enhanced diffusion for Brownian motion on a tilted periodic potential. Expressing the effective diffusion in terms of the eigenvalue band structure, we establish a connection between band gaps in the eigenspectrum and enhanced diffusion. We explain this connection for a simple cosine potential with a linear force and then generalize to more complicated potentials including one-dimensional potentials with multiple frequency components and nonseparable multidimensional potentials. We find that potentials with multiple band gaps in the eigenspectrum can lead to multiple maxima or broadening of the force-diffusion curve. These features are likely to be observable in experiments.

Thermodynamic uncertainty relations and molecular-scale energy conversion.

The thermodynamic uncertainty relation (TUR) is a universal constraint for nonequilibrium steady states that requires the entropy production rate to be greater than the relative magnitude of current fluctuations. It has potentially important implications for the thermodynamic efficiency of molecular-scale energy conversion in both biological and artificial systems. An alternative multidimensional thermodynamic uncertainty relation (MTUR) has also been proposed. In this paper we apply the TUR and the MTUR to a description of molecular-scale energy conversion that explicitly contains the degrees of freedom exchanging energy via a time-independent multidimensional periodic potential. The TUR and the MTUR are found to be universal lower bounds on the entropy generation rate and provide upper bounds on the thermodynamic efficiency. The TUR is found to provide only a weak bound while the MTUR provides a much tighter constraint by taking into account correlations between degrees of freedom. The MTUR is found to provide a tight bound in the near or far from equilibrium regimes but not in the intermediate force regime. Collectively, these results demonstrate that the MTUR is more appropriate than the TUR for energy conversion processes, but that both diverge from the actual entropy generation in certain regimes.

Reconstructing free-energy landscapes for cyclic molecular motors using full multidimensional or partial one-dimensional dynamic information.

Diffusion on a free-energy landscape is a fundamental framework for describing molecular motors. In the landscape framework, energy conversion between different forms of energy, e.g., chemical and mechanical, is explicitly described using multidimensional nonseparable potential landscapes. We present a k-space method for reconstructing multidimensional free-energy landscapes from stochastic single-molecule trajectories. For a variety of two-dimensional model potential landscapes, we demonstrate the robustness of the method by reconstructing the landscapes using full dynamic information, i.e., simulated two-dimensional stochastic trajectories. We then consider the case where the stochastic trajectory is known only along one dimension. With this partial dynamic information, the reconstruction of the full two-dimensional landscape is severely limited in the majority of cases. However, we reconstruct effective one-dimensional landscapes for the two-dimensional model potentials. We discuss the interpretation of the one-dimensional landscapes and identify signatures of energy conversion. Finally, we consider the implications of these results for biological molecular motors experiments.

Analysing single-molecule trajectories to reconstruct free-energy landscapes of cyclic motor proteins.

Stochastic trajectories measured in single-molecule experiments have provided key insights into the microscopic behaviour of cyclic motor proteins. However, the fundamental free-energy landscapes of motor proteins are currently only able to be determined by computationally intensive numerical methods that do not take advantage of available single-trajectory data. In this paper we present a robust method for analysing single-molecule trajectories of cyclic motor proteins to reconstruct their free-energy landscapes. We use simulated trajectories on model potential landscapes to show the reliable reconstruction of the potentials. We determine the accuracy of the reconstruction method for common precision limitations and show that the method converges logarithmically. These results are then used to determine the experimental precision required to reconstruct a potential with a desired accuracy. The key advantages of the method are that it is simple to implement, is free of numerical difficulties that plague existing methods and is easily generalizable to higher dimensions.

Tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor.

We present a tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor. Our derivation is based on the continuous diffusion equation for overdamped Brownian motion on a time-independent tilted periodic potential in two dimensions. The free-energy potential is nonseparable to allow coupling between the chemical and mechanical degrees of freedom. We formally discretize the chemical coordinate by expanding in Wannier states that are localized along the chemical coordinate and parametrized along the mechanical coordinate. A discrete-continuous equation is derived that is valid for anisotropic systems with weak mechanochemical coupling and deep potential wells along the chemical coordinate. The discrete-continuous description is consistent with established theoretical models of molecular motors with discrete chemical states but is constrained by the underlying continuous two-dimensional potential. In particular, we derive analytic expressions for the effective potential along the mechanical coordinate and for the rate of thermal hopping between chemical states. We determine the thermodynamic efficiency of energy conversion and find that, for a molecular motor with one chemical state per cycle, the derived discrete-continuous equation can accurately describe mechanochemical coupling but cannot describe energy conversion.

Reconstructing free-energy landscapes for nonequilibrium periodic potentials.

We present a method for reconstructing the free-energy landscape of overdamped Brownian motion on a tilted periodic potential. Our approach exploits the periodicity of the system by using the k-space form of the Smoluchowski equation and we employ an iterative approach to determine the nonequilibrium tilt. We reconstruct landscapes for a number of example potentials to show the applicability of the method to both deep and shallow wells and near-to- and far-from-equilibrium regimes. The method converges logarithmically with the number of Fourier terms in the potential.

Local discretization method for overdamped Brownian motion on a potential with multiple deep wells.

We present a general method for transforming the continuous diffusion equation describing overdamped Brownian motion on a time-independent potential with multiple deep wells to a discrete master equation. The method is based on an expansion in localized basis states of local metastable potentials that match the full potential in the region of each potential well. Unlike previous basis methods for discretizing Brownian motion on a potential, this approach is valid for periodic potentials with varying multiple deep wells per period and can also be applied to nonperiodic systems. We apply the method to a range of potentials and find that potential wells that are deep compared to five times the thermal energy can be associated with a discrete localized state while shallower wells are better incorporated into the local metastable potentials of neighboring deep potential wells.

Tight-binding approach to overdamped Brownian motion on a bichromatic periodic potential.

We present a theoretical treatment of overdamped Brownian motion on a time-independent bichromatic periodic potential with spatially fast- and slow-changing components. In our approach, we generalize the Wannier basis commonly used in the analysis of periodic systems to define a basis of S states that are localized at local minima of the potential. We demonstrate that the S states are orthonormal and complete on the length scale of the periodicity of the fast-changing potential, and we use the S-state basis to transform the continuous Smoluchowski equation for the system to a discrete master equation describing hopping between local minima. We identify the parameter regime where the master equation description is valid and show that the interwell hopping rates are well approximated by Kramers' escape rate in the limit of deep potential minima. Finally, we use the master equation to explore the system dynamics and determine the drift and diffusion for the system.

Numerical study of the tight-binding approach to overdamped Brownian motion on a tilted periodic potential.

We present a numerical study of the tight-binding approach to overdamped Brownian motion on a tilted periodic potential. In the tight-binding method the probability density is expanded on a basis of Wannier states to transform the Smoluchowski equation to a discrete master equation that can be interpreted in terms of thermal hopping between potential minima. We calculate the Wannier states and hopping rates for a variety of potentials, including tilted cosine and ratchet potentials. For deep potential minima the Wannier states are well localized and the hopping rates between nearest-neighbor states are qualitatively well described by Kramers' escape rate. The next-nearest-neighbor hopping rates are negative and must be negligible compared to the nearest-neighbor rates for the discrete master equation treatment to be valid. We find that the validity of the master equation extends beyond the quantitative applicability of Kramers' escape rate.

Energy transfer in a molecular motor in the Kramers regime.

We present a theoretical treatment of energy transfer in a molecular motor described in terms of overdamped Brownian motion on a multidimensional tilted periodic potential. The tilt represents a thermodynamic force driving the system out of equilibrium and, for nonseparable potentials, energy transfer occurs between degrees of freedom. For deep potential wells, the continuous theory transforms to a discrete master equation that is tractable analytically. We use this master equation to derive formal expressions for the hopping rates, drift and diffusion, and the efficiency and rate of energy transfer in terms of the thermodynamic force. These results span both strong and weak coupling between degrees of freedom, describe the near and far from equilibrium regimes, and are consistent with generalized detailed balance and the Onsager relations. We thereby derive a number of diverse results for molecular motors within a single theoretical framework.

Tight-binding approach to overdamped Brownian motion on a multidimensional tilted periodic potential.

We present a theoretical treatment of overdamped Brownian motion on a multidimensional tilted periodic potential that is analogous to the tight-binding model of quantum mechanics. In our approach, we expand the continuous Smoluchowski equation in the localized Wannier states of the periodic potential to derive a discrete master equation. This master equation can be interpreted in terms of hopping within and between Bloch bands, and for weak tilting and long times we show that a single-band description is valid. In the limit of deep potential wells, we derive a simple functional dependence of the hopping rates and the lowest band eigenvalues on the tilt. We also derive formal expressions for the drift and diffusion in terms of the lowest band eigenvalues.

Thermal fluctuation statistics in a molecular motor described by a multidimensional master equation.

We present a theoretical investigation of thermal fluctuation statistics in a molecular motor. Energy transfer in the motor is described using a multidimensional discrete master equation with nearest-neighbor hopping. In this theory, energy transfer leads to statistical correlations between thermal fluctuations in different degrees of freedom. For long times, the energy transfer is a multivariate diffusion process with constant drift and diffusion. The fluctuations and drift align in the strong-coupling limit enabling a one-dimensional description along the coupled coordinate. We derive formal expressions for the probability distribution and simulate single trajectories of the system in the near- and far-from-equilibrium limits both for strong and weak coupling. Our results show that the hopping statistics provide an opportunity to distinguish different operating regimes.

Bragg scattering of cooper pairs in an ultracold Fermi gas.

We present a theoretical treatment of Bragg scattering of a degenerate Fermi gas in the weakly interacting BCS regime. Our numerical calculations predict correlated scattering of Cooper pairs into a spherical shell in momentum space. The scattered shell of correlated atoms is centered at half the usual Bragg momentum transfer, and can be clearly distinguished from atoms scattered by the usual single-particle Bragg mechanism. We develop an analytic model that explains key features of the correlated-pair Bragg scattering, and determine the dependence of that scattering on the initial pair correlations in the gas.

Impact of a double dose of sulphadoxine-pyrimethamine to reduce prevalence of pregnancy malaria in southern Mozambique.

Malarial infection during pregnancy increases the risks of severe sequelae for the pregnant woman and the risk of delivering a low birthweight baby. The aim of this intervention study was to reduce significantly the prevalence of malaria parasitaemia in adolescent parturients in Matola and Boane in Mozambique. The study was focused upon the most malaria-vulnerable group, adolescent nulliparous and primiparous women. After completing the usual antenatal clinic and giving informed consent, 600 pregnant women were randomly chosen in a double blind manner to one of two regimens comparing the prevailing routine (placebo) for malaria prevention with a two dose regimen of sulphadoxine-pyrimethamine (SP). The first dose was given at enrollment with a second dose at the beginning of the third trimester. At delivery maternal and placental malaria parasitaemia as well as birthweight and gestational duration were analysed. At booking the prevalence of malaria parasitaemia was 35.3% in the placebo group and 30.6% in the SP group. At the second dose, the prevalence of malaria parasitaemia in the placebo group and SP group was 19.7% and 8.7%, respectively. This implies a relative risk (RR) of 2.24 with 95% CI (1.34, 3.75). The corresponding figures at delivery were 13.6% and 6.3% with an RR of 2.22 (1.07, 4.60) and in placenta 13.3% and 2.4% with an RR of 4.87 (1.58, 15.0). Newborns with malaria within 7 days were significantly more frequent in the placebo group, 6.4% and 0.7% respectively, with an RR of 6.55 (1.20, 35.7). Almost all (approximately 98%) of the women studied had Plasmodium falciparum, the remainder had P. malariae and P. ovale. The mean birthweight in the SP group was 3077 g and in the placebo group 2926 g. The estimated mean difference between the two groups was 151 g with 95% CI (51, 252). The mean placental weight in the placebo group was 596 and 645 g in the SP group, implying a difference of 49 g with a 95% CI (11, 88). The mean gestational duration was 6.1 days longer in the SP group, 95% CI (1.5, 10.6). In the placebo group there were two cases of urticaria and one case of nausea; in the SP group there was one case of vomiting. No newborn showed any sign of serious SP side-effect. Two doses of SP were enough to significantly reduce the prevalence of peripheral and placental malaria parasitaemia among young nulliparous and primiparous pregnant women in Matola and Boane.

Perinatal outcome in an obstetric cohort of Mozambican women.

A prospective cohort of 908 consecutively enrolled pregnant women with biparietal diameter (DBP) compatible with gestational age equal to or below 21 weeks were followed up regularly at 2-4 weeks intervals. Normal antenatal care routine was applied. The newborns were followed until 7 days postpartum. The setting was two suburban antenatal clinics in Maputo and the delivery ward at the Maputo Central Hospital. The main outcome variables were low birth weight (LBW), preterm delivery, intrauterine fetal death, perinatal death and small for gestational age (SGA). For each of these variables the odds ratio for maternal risk factors was estimated with 95 per cent confidence interval and multiple logistic regression analysis was used. LBW occurred in 16.2 per cent and low maternal weight, low weight gain during pregnancy and not having a living child were risk factors. Prevalence of preterm birth was 15.4 per cent and low weight gain during pregnancy and malaria in the perinatal period were risk factors. Four per cent of mothers delivered stillborns and syphilis serology (positive VDRL test) was a risk factor. Perinatal death occurred in 4.7 per cent. These deaths were associated with being SGA, LBW or preterm at birth. Of the cohort women, 9.7 per cent delivered SGA newborns. It was concluded that maternal constitutional factors, particularly maternal weight gain, maternal height and maternal weight as well as syphilis and malaria during pregnancy, need to be given attention concerning the adverse outcomes addressed. The establishment of an obstetric cohort, followed prospectively, was possible in a low-income setting with limited numbers lost to follow-up at delivery.

Maternal and fetal characteristics in an obstetric cohort in Mozambique.

A sample of 908 Mozambican pregnant women with gestational age < or = 21 weeks (as measured by ultrasound) were followed fortnightly from their first antenatal clinic visit until the end of the perinatal period. All women attended two suburban/semirural antenatal clinics in Maputo. Only 9% were lost to follow-up. Pre-term delivery occurred in 15.4% of women and low birthweight (LBW) in 16.2%. Mean birthweight was 2.91 kg. Perinatal death occurred in 4.7%. This obstetric cohort provides valuable baseline data to be used as reference. With substantial efforts, the non-compliance with follow-up at birth could be kept at a low level.

An intervention study to reduce adverse pregnancy outcomes as a result of syphilis in Mozambique.

OBJECTIVES: To create and evaluate an alternative screening approach among pregnant women in order to reduce adverse pregnancy outcome as a result of syphilis in Mozambique. METHODS: Four suburban antenatal clinics, two "control" and two "intervention" clinics, were compared regarding syphilis screening and treatment. Pregnant women with positive rapid plasma reagin (RPR) test (n = 929) were enrolled, 453 in the intervention and 476 in the control clinics. In control clinics the normal routine regarding syphilis screening was followed for 383 women remaining for follow up. In intervention clinics nurse midwives were trained to perform the RPR test. RPR seropositive cases were immediately treated on site by the nurse midwives and the partners were invited to come any afternoon for treatment. In the third trimester (around 30 weeks) a new RPR test was performed and all women with positive RPR test results were again treated and the partners were invited to come for treatment. RESULTS: At delivery, the drop out rate was 15.7% in the intervention and 20.1% in the control group. The perinatal mortality was significantly higher in the control group than in the intervention group, 3.4% v 1.3% (p = 0.030). At delivery the intervention group had significantly more negative RPR results--40.9% v 24.2% (p = 0.000). CONCLUSION: More active training of nurse midwives in antenatal care to perform on site RPR tests, to give syphilis treatment, and to notify partners results in improved perinatal outcome and more seronegative parturient women.