Relationship between the Absorption and Emission Spectra and the "Red Drop" in the Action Spectra of Fluorescence In Vivo

The Stepanov equation, relating the intensity of emission, f_e($\bar{v}$), at a given frequency, and that of absorption, k($\bar{v}$), at the same frequency, is applied, in its modified form (see equation 3 in text) to suspensions of Chlorella, Porphyridium, and Anacystis and to chlorophyll solutions. This application can reveal whether the yield of fluorescence, Φ($\bar{v}$), is constant, or changes with frequency. In Chlorella (green alga) a sharp drop of Φ($\bar{v}$) is indicated towards the lower frequencies (longer waves), beginning around $\bar{v}$ = 1.48 × 10⁴cm^-1 (680 mμ); the Φ($\bar{v}$) function calculated from the Stepanov equation is in fair agreement with the directly determined action spectrum for the excitation of chlorophyll fluorescence in this organism. In Porphyridium (red alga) and Anacystis (blue-green alga) application of the Stepanov equation supports the conclusions, derived from direct measurements, of a much earlier “red drop” of the fluorescence excitation spectra. Direct measurements suggest that the drop in Porphyridium may begin at about 1.53 × 10⁴cm^-1 (654 mμ); in Anacystis, it may begin already above 1.57 × 10⁴cm^-1 (<637mμ). These results confirm the relation, postulated earlier by Duysens and others, between the action spectra of photosynthesis and of chlorophyll a fluorescence in algal cells. The relation of these findings to spectroscopic evidence, suggesting the existence of two main chlorophyll a components in vivo, in green as well as in red and blue-green algae, is discussed.     

Comparison of breeding value prediction for two traits in a Nellore-Angus crossbred population using different Bayesian modeling methodologies

The objectives of this study were to 1) compare four models for breeding value prediction using genomic or pedigree information and 2) evaluate the impact of fixed effects that account for family structure. Comparisons were made in a Nellore-Angus population comprising F₂, F₃ and half-siblings to embryo transfer F₂ calves with records for overall temperament at weaning (TEMP; n = 769) and Warner-Bratzler shear force (WBSF; n = 387). After quality control, there were 34,913 whole genome SNP markers remaining. Bayesian methods employed were BayesB ($\tilde{\pi }$ = 0.995 or 0.997 for WBSF or TEMP, respectively) and BayesC (π = 0 and $\tilde{\pi }$), where $\tilde{\pi }$ is the ideal proportion of markers not included. Direct genomic values (DGV) from single trait Bayesian analyses were compared to conventional pedigree-based animal model breeding values. Numerically, BayesC procedures (using $\tilde{\pi }$) had the highest accuracy of all models for WBSF and TEMP ($\widehat{\rho }$_gĝ = 0.843 and 0.923, respectively), but BayesB had the least bias (regression of performance on prediction closest to 1, $\widehat{\beta }$_y,x = 2.886 and 1.755, respectively). Accounting for family structure decreased accuracy and increased bias in prediction of DGV indicating a detrimental impact when used in these prediction methods that simultaneously fit many markers.     

Mixed Venous and Arterial Pco2

The rebreathing method of measuring oxygenated mixed venous Pco₂ (P$\bar{v}$co₂) was originally introduced as a bloodless way to estimate arterial Pco₂ (Paco₂). It has become common practice to subtract 6 mm Hg from the P$\bar{v}$co₂ to obtain the Paco₂ but there are many circumstances in which this leads to an overestimate of the Paco₂. Measurements of P$\bar{v}$co₂ and Paco₂ in 19 patients have shown that a better approximation to Paco₂ under normal conditions of cardiac output and arterial O₂ saturation is Paco₂ = 0·8 P$\bar{v}$co₂. These studies also showed that the P$\bar{v}$co₂ — Paco₂ difference may be much wider, particularly in the presence of arterial unsaturation and a low cardiac output.The factors governing the venoarterial Pco₂ difference are reviewed and their magnitude is calculated to emphasize the complementary roles of measurements of P$\bar{v}$co₂ and Paco₂ in the assessment of patients with cardiorespiratory disease.     

Sediment microbial communities in Great Boiling Spring are controlled by temperature and distinct from water communities

Great Boiling Spring is a large, circumneutral, geothermal spring in the US Great Basin. Twelve samples were collected from water and four different sediment sites on four different dates. Microbial community composition and diversity were assessed by PCR amplification of a portion of the small subunit rRNA gene using a universal primer set followed by pyrosequencing of the V8 region. Analysis of 164 178 quality-filtered pyrotags clearly distinguished sediment and water microbial communities. Water communities were extremely uneven and dominated by the bacterium Thermocrinis. Sediment microbial communities grouped according to temperature and sampling location, with a strong, negative, linear relationship between temperature and richness at all taxonomic levels. Two sediment locations, Site A (87–80 °C) and Site B (79 °C), were predominantly composed of single phylotypes of the bacterial lineage GAL35 ($\widehat{p}$=36.1%), Aeropyrum ($\widehat{p}$=16.6%), the archaeal lineage pSL4 ($\widehat{p}$=15.9%), the archaeal lineage NAG1 ($\widehat{p}$=10.6%) and Thermocrinis ($\widehat{p}$=7.6%). The ammonia-oxidizing archaeon ‘Candidatus Nitrosocaldus'' was relatively abundant in all sediment samples <82 °C ($\widehat{p}$=9.51%), delineating the upper temperature limit for chemolithotrophic ammonia oxidation in this spring. This study underscores the distinctness of water and sediment communities in GBS and the importance of temperature in driving microbial diversity, composition and, ultimately, the functioning of biogeochemical cycles.     

Oxygen uptake,heart rate,perceived exertion,and integrated electromyogram of the lower and upper extremities during level and Nordic walking on a treadmill

The purpose of this study was to characterize responses in oxygen uptake ( V·O2), heart rate (HR), perceived exertion (OMNI scale) and integrated electromyogram (iEMG) readings during incremental Nordic walking (NW) and level walking (LW) on a treadmill. Ten healthy adults (four men, six women), who regularly engaged in physical activity in their daily lives, were enrolled in the study. All subjects were familiar with NW. Each subject began walking at 60 m/min for 3 minutes, with incremental increases of 10 m/min every 2 minutes up to 120 m/min V·O2 , V·E and HR were measured every 30 seconds, and the OMNI scale was used during the final 15 seconds of each exercise. EMG readings were recorded from the triceps brachii, vastus lateralis, biceps femoris, gastrocnemius, and tibialis anterior muscles. V·O2 was significantly higher during NW than during LW, with the exception of the speed of 70 m/min (P < 0.01). V·E and HR were higher during NW than LW at all walking speeds (P < 0.05 to 0.001). OMNI scale of the upper extremities was significantly higher during NW than during LW at all speeds (P < 0.05). Furthermore, the iEMG reading for the VL was lower during NW than during LW at all walking speeds, while the iEMG reading for the BF and GA muscles were significantly lower during NW than LW at some speeds. These data suggest that the use of poles in NW attenuates muscle activity in the lower extremities during the stance and push-off phases, and decreases that of the lower extremities and increase energy expenditure of the upper body and respiratory system at certain walking speeds.     

Higher ventilatory responses during and after passive walking-like leg movement in older individuals

Background

Minute ventilation (V·E) during walking has been shown to be higher in older individuals than in young individuals, but the mechanisms underlying the higher ventilatory response is unclear. Central command and peripheral neural reflex are important neural control mechanisms underlying ventilatory response during exercise. Passive leg movement has been used to exclude the influence of central command due to the lack of voluntary activation of muscles. The aim of the present study was to compare the ventilatory response during and after passive walking-like leg movement (PWM) in young and older individuals.

Methods

Eight young subjects (20 ± 2 years) and seven older subjects (70 ± 1 years) participated in this study. Subjects spent 7 minutes in a quiet standing (QS) position. Thereafter, they performed 14-minute rhythmic PWM at 1 Hz and this was followed by 7 minutes of QS.

Results

V·E values during pre-PWM QS were calculated as 1-minute averages using data obtained between 5 and 6 minutes. V·E values at pre-PWM QS in the young and older groups were 8.4 ± 2.1 and 7.5 ± 1.2 l/minute, respectively. V·E values increased significantly at the first minute of PWM to 11.4 ± 2.2 and 10.4 ± 2.5 l/minute in the young and older groups, respectively (P <0.001). In the young group, V·E at the last minute of PWM (9.2 ± 2.0 l/minute) was not significantly different from that at pre-PWM QS due to a decline in V·E, whereas V·E at the last minute of PWM in the older group (9.4 ± 2.2 l/minute) was still significantly higher (P <0.01). On the other hand, V·E at the first minute of post-PWM QS (7.2 ± 1.8 l/minute) was significantly lower than that during pre-PWM QS in the young group (P <0.05) but not in the older group.

Conclusions

Ventilatory response during and after PWM is higher in older individuals than in young individuals. This may be associated with a mechanism(s) other than central command. Our findings may explain part of the higher V·E response while walking in older individuals.     

Hierarchical Model of Fibrillar Collagen Organization for Interpreting the Second-Order Susceptibility Tensors in Biological Tissue

The second-order nonlinear polarization properties of fibrillar collagen in various rat tissues (vertebrae, tibia, tail tendon, dermis, and cornea) are investigated with polarization-dependent second-harmonic generation (P-SHG) microscopy. Three parameters are extracted: the second-order susceptibility ratio, R = χZZZ(2)′/χZXX(2)′; a measure of the fibril distribution asymmetry, |A|; and the weighted-average fibril orientation, 〈δ〉. A hierarchical organizational model of fibrillar collagen is developed to interpret the second-harmonic generation polarization properties. Highlights of the model include: collagen type (e.g., type-I, type-II), fibril internal structure (e.g., straight, constant-tilt), and fibril architecture (e.g., parallel fibers, intertwined, lamellae). Quantifiable differences in internal structure and architecture of the fibrils are observed. Occurrence histograms of R and |A| distinguished parallel from nonparallel fibril distributions. Parallel distributions possessed low parameter values and variability, whereas nonparallel distributions displayed an increase in values and variability. From the P-SHG parameters of vertebrae tissue, a three-dimensional reconstruction of lamellae of intervertebral disk is presented.     

Enhancement of Chemotactic Cell Aggregation by Haptotactic Cell-To-Cell Interaction

The crawling of biological cell is a complex phenomenon involving various biochemical and mechanical processes. Some of these processes are intrinsic to individual cells, while others pertain to cell-to-cell interactions and to their responses to extrinsically imposed cues. Here, we report an interesting aggregation dynamics of mathematical model cells, when they perform chemotaxis in response to an externally imposed global chemical gradient while they influence each other through a haptotaxis-mediated social interaction, which confers intriguing trail patterns. In the absence of the cell-to-cell interaction, the equilibrium population density profile fits well to that of a simple Keller-Segal population dynamic model, in which a chemotactic current density J→chemo∼∇p competes with a normal diffusive current density J→diff∼∇ρ, where p and ρ refer to the concentration of chemoattractant and population density, respectively. We find that the cell-to-cell interaction confers a far more compact aggregation resulting in a much higher peak equilibrium cell density. The mathematical model system is applicable to many biological systems such as swarming microglia and neutrophils or accumulating ants towards a localized food source.     

Deactivation of a Negative Regulator: A Distinct Signal Transduction Mechanism,Pronounced in Akt Signaling

Kinase cascades, in which enzymes are sequentially activated by phosphorylation, are quintessential signaling pathways. Signal transduction is not always achieved by direct activation, however. Often, kinases activate pathways by deactivation of a negative regulator; this indirect mechanism, pervasive in Akt signaling, has yet to be systematically explored. Here, we show that the indirect mechanism has properties that are distinct from direct activation. With comparable parameters, the indirect mechanism yields a broader range of sensitivity to the input, beyond saturation of regulator phosphorylation, and kinetics that become progressively slower, not faster, with increasing input strength. These properties can be integrated in network motifs to produce desired responses, as in the case of feedforward loops.Phosphorylation of proteins and lipids, catalyzed by specific kinase enzymes, is ubiquitous in intracellular signal transduction. A classic example in eukaryotes is the canonical structure of the mitogen-activated protein kinase cascades, in which three kinases are sequentially activated by phosphorylation (1). Another example is the PI3K (phosphoinositide 3-kinase)/Akt pathway, which (like the mammalian mitogen-activated protein kinases) is prominently dysregulated in human cancers (2). Type-I PI3Ks phosphorylate a lipid substrate to produce the lipid second messenger, PIP₃, which recruits the protein kinase Akt and mediates its activation by phosphorylation (3,4). In no small part because of these important pathways, we typically think of phosphorylation as a direct means of activating molecular interactions and reactions in signal transduction. This is not the only way to increase the flux through a signaling pathway, however. Consider signaling downstream of Akt, which phosphorylates a host of protein substrates to affect diverse functions. A survey of the Akt signaling hub shows that many of these reactions result in a decrease, rather than an increase, in activity/function of the substrates (3). And, among those substrates, the four listed in Fig. S1 in the Supporting Material). Whereas negative regulators are appreciated for their roles in feedback adaptation of signaling, the implications of deactivating a negative regulator as an indirect mechanism of pathway activation has yet to be explored.

Table 1

Survey of Akt substrates and downstream signaling

Length-Based Assessment of Coral Reef Fish Populations in the Main and Northwestern Hawaiian Islands

The coral reef fish community of Hawaii is composed of hundreds of species, supports a multimillion dollar fishing and tourism industry, and is of great cultural importance to the local population. However, a major stock assessment of Hawaiian coral reef fish populations has not yet been conducted. Here we used the robust indicator variable “average length in the exploited phase of the population (L¯)”, estimated from size composition data from commercial fisheries trip reports and fishery-independent diver surveys, to evaluate exploitation rates for 19 Hawaiian reef fishes. By and large, the average lengths obtained from diver surveys agreed well with those from commercial data. We used the estimated exploitation rates coupled with life history parameters synthesized from the literature to parameterize a numerical population model and generate stock sustainability metrics such as spawning potential ratios (SPR). We found good agreement between predicted average lengths in an unfished population (from our population model) and those observed from diver surveys in the largely unexploited Northwestern Hawaiian Islands. Of 19 exploited reef fish species assessed in the main Hawaiian Islands, 9 had SPRs close to or below the 30% overfishing threshold. In general, longer-lived species such as surgeonfishes, the redlip parrotfish (Scarus rubroviolaceus), and the gray snapper (Aprion virescens) had the lowest SPRs, while short-lived species such as goatfishes and jacks, as well as two invasive species (Lutjanus kasmira and Cephalopholis argus), had SPRs above the 30% threshold.     

Relationship between plasma asymmetric dimethylarginine and nitric oxide levels affects aerobic exercise training-induced reduction of arterial stiffness in middle-aged and older adults

[Purpose]Aerobic exercise training (AT) reverses aging-induced deterioration of arterial stiffness via increased arterial nitric oxide (NO) production. Asymmetric dimethylarginine (ADMA), an endogenous inhibitor of NO synthase, was decreased by AT. However, whether AT-induced changes in ADMA levels are related to changes in nitrite/nitrate (NOx) levels remains unclear. Accordingly, we aimed to clarify whether the relationship between plasma ADMA and NOx levels affected the AT-induced reduction of arterial stiffness in middle-aged and older adults.[Methods]Thirty-one healthy middle-aged and older male and female subjects (66.4 ± 1.3 years) were randomly divided into two groups: exercise intervention and sedentary controls. Subjects in the training group completed an 8-week AT (60%–70% peak oxygen uptake [V˙O2peak] for 45 min, 3 days/week).[Results]AT significantly increased V˙O2peak (p < 0.05) and decreased carotid β-stiffness (p < 0.01). Moreover, plasma ADMA levels were significantly decreased while plasma NOx levels and NOx/ADMA ratio were significantly increased by AT (p < 0.01). Additionally, no sex differences in AT-induced changes of circulating ADMA and NOx levels, NOx/ADMA ratio, and carotid β-stiffness were observed. Furthermore, the AT-induced increase in circulating ADMA levels was negatively correlated with an increase in circulating NOx levels (r = -0.414, p < 0.05), and the AT-induced increase in NOx/ADMA ratio was negatively correlated with a decrease in carotid β-stiffness (r = -0.514, p < 0.01).[Conclusion]These results suggest that the increase in circulating NOx with reduction of ADMA elicited by AT is associated with a decrease in arterial stiffness regardless of sex in middle-aged and older adults.     

Heuristic estimation of the α/β ratio for a cohort of Mexican patients with prostate cancer treated with external radiotherapy techniques

BackgroundThe aim of the study was to Estimate and compare the radiobiological ratio α/β with the heuristic method for a cohort of Mexican patients with prostate cancer (PCa) who were treated with external radiotherapy (RT) techniques at three Hospital Institutions in Mexico City. With the Kaplan-Meier technique and the Cox proportional hazards model, the biochemical relapse-free survival (bRFS) is determined and characterized for cohorts of Mexican patients with PCa who received treatment with external RT. Using these clinical outcomes, the radiobiological parameter α/β is determined using the heuristic methodology of Pedicini et. al.Materials and methodsThe α/β is calculated from the survival curves for different treatment schemes implemented at three distinct hospitals. The Pedicini’s techniques allow to determine the parameters α/β, k and N₀ when treatments are not radiobiologically equivalent, therefore, are built up of a set of curved pairs for the biologically effective dose (BED) versus the ratio α/β, where the ratio is given by the intersection for each pair of curves.ResultsSix different values of α/β were found: the first α/β = 2.46 Gy, the second α/β = 3.30 Gy, the third for α/β = 3.25 Gy, the fourth α/β = 3.24 Gy, the fifth α/β = 3.38 Gy and the last α/β = 4.08 Gy. These values can be explained as follows: a) The bRFS of the schemes presents a statistical variation; b) The absorbed doses given to the patient present uncertainties on the physical dosimetry that are not on the modeling; c) Finally, in the model for the bRFS of Eq. (3), there are parameters that have to be considered, such as: the number of clonogenic tumor cells N₀, the overall treatment time (OTT), the kick-off time for tumor repopulation T_k and the repopulation doubling time. Therefore, the mean value to α/β for all schemes has an average value of 3.29 (± 0.52) Gy.ConclusionsThe value of α/β¯=3.29 (±0.52) Gy is determined from cohorts of Mexican patients with PC a treated with external radiotherapy using the time-dependent LQ model, which is a higher value with respect to the “dogma” value of α/β 1.5 Gy obtained with the LQ model without temporal dependence. Therefore, there is a possibility of optimizing treatments radiobiologically and improving the results of bRFS in Mexican patients with PCa treated with external radiotherapy.     

Rhinos in the Parks: An Island-Wide Survey of the Last Wild Population of the Sumatran Rhinoceros

In the 200 years since the Sumatran rhinoceros was first scientifically described (Fisher 1814), the range of the species has contracted from a broad region in Southeast Asia to three areas on the island of Sumatra and one in Kalimantan, Indonesia. Assessing population and spatial distribution of this very rare species is challenging because of their elusiveness and very low population number. Using an occupancy model with spatial dependency, we assessed the fraction of the total landscape occupied by Sumatran rhinos over a 30,345-km² survey area and the effects of covariates in the areas where they are known to occur. In the Leuser Landscape (surveyed in 2007), the model averaging result of conditional occupancy estimate was ψ^(SE[ψ^])=0.151(0.109) or 2,371.47 km², and the model averaging result of replicated level detection probability p^(SE[p^])=0.252(0.267); in Way Kambas National Park—2008: ψ^(SE[ψ^])=0.468(0.165) or 634.18 km², and p^(SE[p^])=0.138(0.571); and in Bukit Barisan Selatan National Park—2010: ψ^(SE[ψ^])=0.322(0.049) or 819.67 km², and p^(SE[p^])=0.365(0.42). In the Leuser Landscape, rhino occurrence was positively associated with primary dry land forest and rivers, and negatively associated with the presence of a road. In Way Kambas, occurrence was negatively associated with the presence of a road. In Bukit Barisan Selatan, occurrence was negatively associated with presence of primary dryland forest and rivers. Using the probabilities of site occupancy, we developed spatially explicit maps that can be used to outline intensive protection zones for in-situ conservation efforts, and provide a detailed assessment of conserving Sumatran rhinos in the wild. We summarize our core recommendation in four points: consolidate small population, strong protection, determine the percentage of breeding females, and recognize the cost of doing nothing. To reduce the probability of poaching, here we present only the randomized location of site level occupancy in our result while retaining the overall estimation of occupancy for a given area.     

Angle of Inclination of Tank-Treading Red Cells: Dependence on Shear Rate and Suspending Medium

Red cells suspended in solutions much more viscous than blood plasma assume an almost steady-state orientation when sheared above a threshold value of shear rate. This orientation is a consequence of the motion of the membrane around the red cell called tank-treading. Observed along the undisturbed vorticity of the shear flow, tank-treading red cells appear as slender bodies. Their orientation can be quantified as an angle of inclination (θ) of the major axis with respect to the undisturbed flow direction. We measured θ using solution viscosities (η₀) and shear rates (γ˙) covering one and three orders of magnitude, respectively. At the lower values of η₀, θ was almost independent of γ˙. At the higher values of η₀, θ displayed a maximum at intermediate shear rates. The respective maximal values of θ increased by ∼10° from 10.7 to 104 mPas. After accounting for the absent membrane viscosity in models by using an increased cytoplasmic viscosity, their predictions of θ agree qualitatively with our data. Comparison of the observed variation of θ at constant γ˙ with model results suggests a change in the reference configuration of the shear stiffness of the membrane.     

A Novel TetR-Regulating Peptide Turns off rtTA-Mediated Activation of Gene Expression

Conditional regulation of gene expression is a powerful and indispensable method for analyzing gene function. The “Tet-On” system is a tool widely used for that purpose. Here, the transregulator rtTA mediates expression of a gene of interest after addition of the small molecule effector doxycycline. Although very effective in rapidly turning on gene expression, the system is hampered by the long half-life of doxycycline which makes shutting down gene expression rapidly very difficult to achieve. We isolated an rtTA-binding peptide by in vivo selection that acts as a doxycycline antagonist and leads to rapid and efficient shut down of rtTA-mediated reporter gene expression in a human cell line. This peptide represents the basis for novel effector molecules which complement the “Tet-system” by enabling the investigator to rapidly turn gene expression not just on at will, but now also off.     

The gating charge should not be estimated by fitting a two-state model to a Q-V curve

The voltage dependence of charges in voltage-sensitive proteins, typically displayed as charge versus voltage (Q-V) curves, is often quantified by fitting it to a simple two-state Boltzmann function. This procedure overlooks the fact that the fitted parameters, including the total charge, may be incorrect if the charge is moving in multiple steps. We present here the derivation of a general formulation for Q-V curves from multistate sequential models, including the case of infinite number of states. We demonstrate that the commonly used method to estimate the charge per molecule using a simple Boltzmann fit is not only inadequate, but in most cases, it underestimates the moving charge times the fraction of the field.Many ion channels, transporters, enzymes, receptors, and pumps are voltage dependent. This voltage dependence is the result of voltage-induced translocation of intrinsic charges that, in some way, affects the conformation of the molecule. The movement of such charges is manifested as a current that can be recorded under voltage clamp. The best-known examples of these currents are “gating” currents in voltage-gated channels and “sensing” currents in voltage-sensitive phosphatases. The time integral of the gating or sensing current as a function of voltage (V) is the displaced charge Q(V), normally called the Q-V curve.It is important to estimate how much is the total amount of net charge per molecule (Q_max) that relocates within the electric field because it determines whether a small or a large change in voltage is necessary to affect the function of the protein. Most importantly, knowing Q_max is critical if one wishes to correlate charge movement with structural changes in the protein. The charge is the time integral of the current, and it corresponds to the product of the actual moving charge times the fraction of the field it traverses. Therefore, correlating charge movement with structure requires knowledge of where the charged groups are located and the electric field profile. In recent papers by Chowdhury and Chanda (2012) and Sigg (2013), it was demonstrated that the total energy of activating the voltage sensor is equal to Q_max V_M, where V_M is the median voltage of charge transfer, a value that is only equal to the half-point of activation V_1/2 for symmetrical Q-V curves. V_M is easily estimated from the Q-V curve, but Q_max must be obtained with other methods because, as we will show here, it is not directly derived from the Q-V curve in the general case.The typical methods used to estimate charge per molecule Q_max include measurements of limiting slope (Almers, 1978) and the ratio of total charge divided by the number of molecules (Schoppa et al., 1992). The discussion on implementation, accuracy, and reliability of these methodologies has been addressed many times in the literature, and it will not be discussed here (see Sigg and Bezanilla, 1997). However, it is worth mentioning that these approaches tend to be technically demanding, thus driving researchers to seek alternative avenues toward estimating the total charge per molecule. Particularly, we will discuss here the use of a two-state Boltzmann distribution for this purpose. Our intention is to demonstrate that this commonly used method to estimate the charge per molecule is generally incorrect and likely to give a lower bound of the moving charge times the fraction of the field.The two-state Boltzmann distribution describes a charged particle that can only be in one of two positions or states that we could call S₁ and S₂. When the particle with charge Q_max (in units of electronic charge) moves from S₁ to S₂, or vice versa, it does it in a single step. The average charge found in position S₂, Q(V), will depend on the energy difference between S₁ and S₂, and the charge of the particle. The equation that describes Q(V) is:Q(V)=Qmax1+exp[−Qmax(V−V1/2)kT],(1)where V_1/2 is the potential at which the charge is equally distributed between S₁ and S₂, and k and T are the Boltzmann constant and absolute temperature, respectively. The Q(V) is typically normalized by dividing Eq. 1 by the total charge Q_max. The resulting function is frequently called a “single Boltzmann” in the literature and is used to fit normalized, experimentally obtained Q-V curves. The fit yields an apparent V_1/2 (V∼1/2) and an apparent Q_MAX (Q−max), and this last value is then attributed to be the total charge moving Q_max. Indeed, this is correct but only for the case of a charge moving between two positions in a single step. However, the value of Q−max thus obtained does not represent the charge per molecule for the more general (and frequent) case when the charge moves in more than one step.To demonstrate the above statement and also estimate the possible error in using the fitted Q−max from Eq. 1, let us consider the case when the gating charge moves in a series of n steps between n + 1 states, each step with a fractional charge z_i (in units of electronic charge e₀) that will add up to the total charge Q_max.S1↔μ1S2↔μ2…Si↔μiSi+1…Sn↔μnSn+1The probability of being in each of the states S_i is labeled as P_i, and the equilibrium constant of each step is given byμi=exp[zi(V−Vi)kT], i=1…n,where z_i is the charge (in units of e₀) of step i, and V_i is the membrane potential that makes the equilibrium constant equal 1. In steady state, the solution of P_i can be obtained by combiningPi+1Pi=μi, i=1…n and ∑i=1i=n+1Pi=1,givingPi+1=∏m=1iμm1+∑j=1n∏k=1jμk, i=1…n and P1=11+∑j=1n∏k=1jμk.We define the reaction coordinate along the moved charged q asqi=∑j=1izj, i=1…n.The Q-V curve is defined asQ(V)=∑i=1nqiPi+1.Then, replacing P_i yieldsQ(V)=∑i=1n[∑j=1izj][∏m=1iμm]1+∑j=1n∏k=1jμk,or written explicitly as a function of V:Q(V)=∑i=1n[∑j=1izj][∏m=1iexp[zm(V−Vm)kT]]1+∑j=1n∏k=1jexp[zk(V−Vk)kT].(2)Eq. 2 is a general solution of a sequential model with n + 1 states with arbitrary valences and V_i’s for each transition. We can easily see that Eq. 2 has a very different form than Eq. 1, except when there is only a single transition (n = 1). In this latter case, Eq. 2 reduces to Eq. 1 because z₁ and V₁ are equal to Q_max and V_1/2, respectively. For the more general situation where n > 1, if one fits the Q(V) relation obeying Eq. 2 with Eq. 1, the fitted Q−max value will not correspond to the sum of the z_i values (see examples below and Fig. 1). A simple way to visualize the discrepancy between the predicted value of Eqs. 1 and 2 is to compute the maximum slope of the Q-V curve. This can be done analytically assuming that V_i = V_o for all transitions and that the total charge Q_max is evenly divided among those transitions. The limit of the first derivative of the Q(V) with respect to V evaluated at V = V_o is given by this equation:dQ(V)dV|V=V0=Qmax(n+2)12nkT.(3)From Eq. 3, it can be seen that the slope of the Q-V curve decreases with the number of transitions being maximum and equal to Q_max /(4kT) when n = 1 (two states) and a minimum equal to Q_max /(12kT) when n goes to infinity, which is the continuous case (see next paragraph).Open in a separate windowFigure 1.Examples of normalized Q-V curves for a Q_max = 4 computed with Eq. 2 for the cases of one, two, three, four, and six transitions and the continuous case using Eq. 5 (squares). All the Q-V curves were fitted with Eq. 1 (lines). The insets show the fitted valence (Q−max) and half-point (V∼1/2).

Infinite number of steps

Eq. 2 can be generalized to the case where the charge moves continuously, corresponding to an infinite number of steps. If we makez_i = Q_max/n, ?i = 1…n, ??V_i = V_o, ?i = 1…n, then all µ_i = µ, and we can write Eq. 2 as the normalized Q(V) in the limit when n goes to infinity:Qnor(V)=limn→∞∑i=1n[∑j=1iQmaxn]∏m=1iexp[Qmax(V−Vo)nkT]Qmax[1+∑i=1n∏j=1iexp[Qmax(V−Vo)nkT]]=[Qmax(V−Vo)−kT]exp[Qmax(V−Vo)kT]+kTQmax(V−Vo)[exp[Qmax(V−Vo)kT]−1].(4)Eq. 4 can also be written asQnor(V)=12[1+coth[Qmax(V−Vo)2kT]−2kTQmax(V−V0)],(5)which is of the same form of the classical equation of paramagnetism (see Kittel, 2005).

Examples

We will illustrate now that data generated by Eq. 2 can be fitted quite well by Eq. 1, thus leading to an incorrect estimate of the total charge moved. Typically, the experimental value of the charge plotted is normalized to its maximum because there is no knowledge of the absolute amount of charge per molecule and the number of molecules. The normalized Q-V curve, Q_nor, is obtained by dividing Q(V) by the sum of all the partial charges.Fig. 1 shows Q_nor computed using Eq. 2 for one, two, three, four, and six transitions and for the continuous case using Eq. 5 (squares) with superimposed fits to a two-state Boltzmann distribution (Eq. 1, lines). The computations were done with equal charge in each step (for a total charge Q_max = 4e₀) and also the same V_i = −25 mV value for all the steps. It is clear that fits are quite acceptable for cases up to four transitions, but the fit significantly deviates in the continuous case.Considering that experimental data normally have significant scatter, it is then quite likely that the experimenter will accept the single-transition fit even for cases where there are six or more transitions (see Fig. 1). In general, the case up to four transitions will look as a very good fit, and the fitted Q−max value may be inaccurately taken and the total charge transported might be underestimated. To illustrate how bad the estimate can be for these cases, we have included as insets the fitted value of Q−max for the cases presented in Fig. 1. It is clear that the estimated value can be as low as a fourth of the real total charge. The estimated value of V∼1/2 is very close to the correct value for all cases, but we have only considered cases in which all V_i’s are the same.It should be noted that if µ_i of the rightmost transition is heavily biased to the last state (V_i is very negative), then the Q−max estimated by fitting a two-state model is much closer to the total gating charge. In a three-state model, it can be shown that the fitted value is exact when V₁→∞ and V₂→−∞ because in that case, it converts into a two-state model. Although these values of V are unrealistic, the fitted value of Q−max can be very close to the total charge when V₂ is much more negative than V₁ (that is, V₁ >> V₂). On the other hand, If V₁ << V₂, the Q-V curve will exhibit a plateau region and, as the difference between V₁ and V₂ decreases, the plateau becomes less obvious and the curve looks monotonic. These cases have been discussed in detail for the two-transition model in Lacroix et al. (2012).We conclude that it is not possible to estimate unequivocally the gating charge per sensor from a “single-Boltzmann” fit to a Q-V curve of a charge moving in multiple transitions. The estimated Q−max value will be a low estimate of the gating charge Q_max, except in the case of the two-state model or the case of a heavily biased late step, which are rare occurrences. It is then safer to call “apparent gating charge” the fitted Q−max value of the single-Boltzmann fit.

Addendum

The most general case in which transitions between states include loops, branches, and steps can be derived directly from the partition function and follows the general thermodynamic treatment by Sigg and Bezanilla (1997), Chowdhury and Chanda (2012), and Sigg (2013). The reaction coordinate is the charge moving in the general case where it evolves from q = 0 to q = Q_max by means of steps, loops, or branches. In that case, the partition function is given byZ=∑iexp(qi(V−Vi)kT).(6)We can compute the mean gating charge, also called the Q-V curve, asQ(V)=〈q〉=kTZ’Z=kTdlnZdV=∑iqiexp(qi(V−Vi)kT)∑iexp(qi(V−Vi)kT).(7)The slope of the Q-V is obtained by taking the derivative of 〈q〉 with respect to V:dQ(V)dV=(kT)2d2lnZdV2.(8)Let us now consider the gating charge fluctuation. The charge fluctuation will depend on the number of possible conformations of the charge and is expected to be a maximum when there are only two possible charged states to dwell. As the number of intermediate states increases, the charge fluctuation decreases. Now, a measure of the charge fluctuation is given by the variance of the gating charge, which can be computed from the partition function as:〈Δq2〉=〈q2〉−〈q〉2=(kT)2(Z’’Z−(Z’Z)2)=(kT)2d2lnZdV2.(9)But the variance (Eq. 9) is identical to the slope of Q(V) (Eq. 8). This implies that the slope of the Q-V is maximum when there are only two states.