Plasma Levels of Middle Molecules to Estimate Residual Kidney Function in Haemodialysis without Urine Collection

Background

Residual Kidney Function (RKF) is associated with survival benefits in haemodialysis (HD) but is difficult to measure without urine collection. Middle molecules such as Cystatin C and β2-microglobulin accumulate in renal disease and plasma levels have been used to estimate kidney function early in this condition. We investigated their use to estimate RKF in patients on HD.

Design

Cystatin C, β2-microglobulin, urea and creatinine levels were studied in patients on incremental high-flux HD or hemodiafiltration(HDF). Over sequential HD sessions, blood was sampled pre- and post-session 1 and pre-session 2, for estimation of these parameters. Urine was collected during the whole interdialytic interval, for estimation of residual GFR (GFR_Residual = mean of urea and creatinine clearance). The relationships of plasma Cystatin C and β2-microglobulin levels to GFR_Residual and urea clearance were determined.

Results

Of the 341 patients studied, 64% had urine output>100ml/day, 32.6% were on high-flux HD and 67.4% on HDF. Parameters most closely correlated with GFR_Residual were 1/β2-micoglobulin (r² 0.67) and 1/Cystatin C (r² 0.50). Both these relationships were weaker at low GFR_Residual. The best regression model for GFR_Residual, explaining 67% of the variation, was: GFRResidual=160.3⋅(1β2m)−4.2 Where β2m is the pre-dialysis β2 microglobulin concentration (mg/L). This model was validated in a separate cohort of 50 patients using Bland-Altman analysis. Areas under the curve in Receiver Operating Characteristic analysis aimed at identifying subjects with urea clearance≥2ml/min/1.73m² was 0.91 for β2-microglobulin and 0.86 for Cystatin C. A plasma β2-microglobulin cut-off of ≤19.2mg/L allowed identification of patients with urea clearance ≥2ml/min/1.73m² with 90% specificity and 65% sensitivity.

Conclusion

Plasma pre-dialysis β2-microglobulin levels can provide estimates of RKF which may have clinical utility and appear superior to cystatin C. Use of cut-off levels to identify patients with RKF may provide a simple way to individualise dialysis dose based on RKF.     

Kinetics and Energetics of Biomolecular Folding and Binding

The ability of biomolecules to fold and to bind to other molecules is fundamental to virtually every living process. Advanced experimental techniques can now reveal how single biomolecules fold or bind against mechanical force, with the force serving as both the regulator and the probe of folding and binding transitions. Here, we present analytical expressions suitable for fitting the major experimental outputs from such experiments to enable their analysis and interpretation. The fit yields the key determinants of the folding and binding processes: the intrinsic on-rate and the location and height of the activation barrier.Dynamic processes in living cells are regulated through conformational changes in biomolecules—their folding into a particular shape or binding to selected partners. The ability of biomolecules to fold and to bind enables them to act as switches, assembly factors, pumps, or force- and displacement-generating motors (1). Folding and binding transitions are often hindered by a free energy barrier. Overcoming the barrier requires energy-demanding rearrangements such as displacing water from the sites of native contacts and breaking nonnative electrostatic contacts, as well as loss of configurational entropy. Once the barrier is crossed, the folded and bound states are stabilized by short-range interactions: hydrogen bonds, favorable hydrophobic effects, and electrostatic and van der Waals attractions (2).Mechanistic information about folding and binding processes is detailed in the folding and binding trajectories of individual molecules: observing an ensemble of molecules may obscure the inherent heterogeneity of these processes. Single-molecule trajectories can be induced, and monitored, by applying force to unfold/unbind a molecule and then relaxing the force until folding or binding is observed (3–5) (Fig. 1). Varying the force relaxation rate shifts the range of forces at which folding or binding occurs, thus broadening the explorable spectrum of molecular responses to force and revealing conformational changes that are otherwise too fast to detect. The measured force-dependent kinetics elucidates the role of force in physiological processes (6) and provides ways to control the timescales, and even the fate, of these processes. The force-dependent data also provides a route to understanding folding and binding in the absence of force—by extrapolating the data to zero force via a fit to a theory.Open in a separate windowFigure 1Schematic of the output from a force-relaxation experiment. The applied force is continuously relaxed from the initial value F₀ until the biomolecule folds or binds, as signified by a sharp increase in the measured force. From multiple repeats of this experiment, distributions of the folding or binding forces are collected (inset). Fitting the force distributions with the derived analytical expression yields the key parameters that determine the kinetics and energetics of folding or binding.In this letter, we derive an analytical expression for the distribution of transition forces, the major output of force-relaxation experiments that probe folding and binding processes. The expression extracts the key determinants of these processes: the on-rate and activation barrier in the absence of force. The theory is first developed in the context of biomolecular folding, and is then extended to cover the binding of a ligand tethered to a receptor. In contrast to unfolding and unbinding, the reverse processes of folding and binding require a theory that accounts for the compliance of the unfolded state, as well as the effect of the tether, to recover the true kinetic parameters of the biomolecule of interest.In a force-relaxation experiment, an unfolded biomolecule or unbound ligand-receptor complex is subject to a stretching force, which is decreased from the initial value F₀ as the pulling device approaches the sample at speed V until a folding or binding transition is observed (Fig. 1) (3–5). Define S(t) as the probability that the molecule has not yet escaped from the unfolded (implied: or unbound) state at time t. When escape is limited by one dominant barrier, S(t) follows the first-order rate equationS˙(t)≡dS(t)dt=−k←(F(t))S(t),where k_←(F(t)) is the on-rate at force F at time t. Because, prior to the transition, the applied force decreases monotonically with time, the distribution of transition forces, p(F), is related to S(t) through p(F)dF=S˙(t)dt, yieldingp(F)=−k←(F)F˙(F)e−∫F0Fk←(F′)F˙(F′)dF′.(1)Here F˙(F)≡dF(t)/dt<0 is the force relaxation rate. The proper normalization of p(F) is readily confirmed by integrating Eq. 1 from the initial force F₀ to negative infinity, the latter accounting for transitions that do not occur by the end of the experiment. Note that the expression for the distribution of folding/binding forces in Eq. 1 differs from its analog for the unfolding process (7) by the limits of integration and a negative sign, reflecting the property of a relaxation experiment to decrease the survival probability S(t) by decreasing the force. Converting the formal expression in Eq. 1 into a form suitable for fitting experimental data requires establishing functional forms for k_←(F) and F˙(F) and analytically solving the integral. These steps are accomplished below.The on-rate k_←(F) is computed by treating the conformational dynamics of the molecule as a random walk on the combined free energy profile G(x,t) = G₀(x) + G_pull(x,t) along the molecular extension x. Here G₀(x) is the intrinsic molecular potential and G_pull(x,t) is the potential of the pulling device. When G(x,t) features a high barrier on the scale of k_BT (k_B is the Boltzmann constant and T the temperature), the dynamics can be treated as diffusive. The unfolded region of the intrinsic potential for a folding process, unlike that for a barrierless process (8), can be captured by the functionG0(x)=ΔG‡ν1−ν(xx‡)11−ν−ΔG‡ν(xx‡),which has a sharp (if ν = 1/2, Fig. 2, inset) or smooth (if ν = 2/3) barrier of height ΔG^‡ and location x^‡. The potential of a pulling device of stiffness κ_S is G_pull(x,t) = κ_S/2(X₀ – Vt – x)² with an initial minimum at X₀ (corresponding to F₀). Applying Kramers formalism (9) to the combined potential G(x,t), we establish the analytical form of the on-rate at force F(t),k←(F)=k0(1+κSκU(F))1ν−12(1+νFx‡ΔG‡)1ν−1×eβΔG‡[1−(1+κSκU(F))2ν1−ν−1(1+νFx‡ΔG‡)1ν],where k₀ is the intrinsic on-rate, β ≡ (k_BT)⁻¹, andκU(F)=ν(1−ν)2ΔG‡x‡2(1+νFx‡ΔG‡)2−1νis the stiffness of the unfolded biomolecule under force F (see the Supporting Material for details on all derivations). The full nonlinear form of G_pull(x,t) was necessary in the derivation because, in contrast to the typically stiff folded state, the unfolded state may be soft (to be exact, 1/2κ_S x^‡2(F) << k_BT may not be satisfied) and thus easily deformed by the pulling device. Because of this deformation, the folding transition faces an extra contribution (regulated by the ratio κ_S/κ_U(F)) to the barrier height, typically negligible for unfolding, that decreases the on-rate in addition to the applied force F.Open in a separate windowFigure 2Contributions to the free energy profile for folding (inset) and binding (main figure). The derived expression (Eq. 2) extracts the on-rate and the location and height of the activation barrier to folding. When applied to binding data, the expression extracts the parameters of the ligand-tether-receptor (LTR) potential G˜0 (x); the proposed algorithm (Eqs. 3 and 4) removes the contribution of the tether potential G_teth(x) to recover the parameters of the intrinsic ligand-receptor (LR) potential G₀(x).The last piece required for Eq. 1, the loading rate F˙(F), is computed as the time derivative of the force F(t) on the unfolded molecule at its most probable extension at time t:F˙(F)=−κSV1+κS/κU(F).Finally, we realize that the integral in Eq. 1 can be solved analytically exactly, both for ν = 1/2 and ν = 2/3, resulting in the analytical expression for the distribution of folding forces:p(F)=k←(F)|F˙(F)|e−k←(F)β|F˙(F)|x‡(1+κSκU(F))νν−1(1+νFx‡ΔG‡)1−1ν.(2)Equation 2 can be readily applied to (normalized) histograms from force-relaxation experiments to extract the parameters of the intrinsic kinetics and energetics of folding. Being exact for ν = 1/2 and ν = 2/3, Eq. 2 is also an accurate approximation for any ν in the interval 1/2 < ν < 2/3 as long as κ_S ≲ κ_U (F) (see Fig. S1 in the Supporting Material). For simplicity, in Eq. 2 we have omitted the term containing F₀ as negligible if F₀ is large enough to prevent folding events.The solution in Eq. 2 reveals properties of the distribution of folding forces that distinguish it from its unfolding counterpart (7):

• 1.The distribution has a positive skew (Fig. 3), as intuitively expected: the rare folding events occur at high forces when the barrier is still high.Open in a separate windowFigure 3Force histograms from folding (left) and binding (right) simulations at several values of the force-relaxation speed (in nanometers per second, indicated at each histogram). Fitting the histograms with the analytical expression in Eq. 2 (lines) recovers the on-rate and activation barrier for folding or binding (2.Increasing the relaxation speed shifts the distribution to lower forces (Fig. 3): faster force relaxation leaves less time for thermal fluctuations to push the system over a high barrier, causing transitions to occur later (i.e., at lower forces), when the barrier is lower.
• 3.The stiffness κ_S and speed V enter Eq. 2 separately, providing independent routes to control the range of folding forces and thus enhance the robustness of a fit.

The application of the above framework to binding experiments on a ligand and receptor connected by a tether (3) involves an additional step—decoupling the effect of the tether—to reconstruct the parameters of ligand-receptor binding. Indeed, the parameters extracted from a fit of experimental histograms to Eq. 2 characterize the ligand-tether-receptor (LTR) potential (k˜0, x˜‡, ΔG˜‡, ν) (Fig. 2). The parameters of the natural ligand-receptor (LR) potential (k₀, x^‡, ΔG^‡) can be recovered using three characteristics of the tether: contour length L; persistence length p; and extension Δℓ of the tether along the direction of the force in the LTR transition state. The values of L and p can be determined from the force-extension curve of the tether (10); these define the tether potential G_teth(x) (Fig. 2). The value of Δℓ can be found from an unbinding experiment (7) on LTR and the geometry of the tether attachment points (see Fig. S3). Approximating the region of the LR potential between the transition and unbound states as harmonic, with no assumptions about the shape of the potential beyond x^‡, the ligand-receptor barrier parameters are thenx‡=α−1α−2x˜‡,ΔG‡=(α−1)22(α−2)x˜‡Fteth(Δℓ+x˜‡),(3)and the intrinsic unimolecular association rate isk0≈k˜0(βΔG‡)32(βΔG˜‡)1ν−12(x˜‡x‡)2eβ(ΔG˜‡−ΔG‡).(4)Here, the force value Fteth(Δℓ+x˜‡) is extracted from the force-extension curve of the tether at extension Δℓ+x˜‡ andα=2(ΔG˜‡−Gteth(Δℓ)+Gteth(Δℓ+x˜‡))x˜‡Fteth(Δℓ+x˜‡),where G_teth(x) is the wormlike-chain potential (see Eq. S13 in the Supporting Material). Equations 3–4 confirm that a tether decreases the height and width of the barrier (see Fig. 2), thus increasing the on-rate.In Fig. 3, the developed analytical framework is applied to folding and binding force histograms from Brownian dynamics simulations at parameters similar to those in the analogous experimental and computational studies (3,5,11) (for details on simulations and fitting procedure, see the Supporting Material). For the stringency of the test, the simulations account for the wormlike-chain nature of the molecular unfolded and LTR unbound states that is not explicitly accounted for in the theory. With optimized binning (12) of the histograms and a least-squares fit, Eqs. 2–4 recover the on-rate, the location and the height of the activation barrier, and the value of ν that best captures how the kinetics scale with force (

1.Multiple relaxation speeds,

2.Folding/binding events at low forces, and

3.A large number of events at each speed.

Table 1

On-rate and the location and height of the activation barrier from the fit of simulated data to the theory in Eq. 2

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

The limitations,dangers, and benefits of simple methods for testing identifiability

In their Commentary paper, Villaverde and Massonis (On testing structural identifiability by a simple scaling method: relying on scaling symmetries can be misleading) have commented on our paper in which we proposed a simple scaling method to test structural identifiability. Our scaling invariance method (SIM) tests for scaling symmetries only, and Villaverde and Massonis correctly show the SIM may fail to detect identifiability problems when a model has other types of symmetries. We agree with the limitations raised by these authors but, also, we emphasize that the method is still valuable for its applicability to a wide variety of models, its simplicity, and even as a tool to introduce the problem of identifiability to investigators with little training in mathematics.

In their Commentary paper, Villaverde and Massonis (On testing structural identifiability by a simple scaling method: relying on scaling symmetries can be misleading [1]) have commented on our paper in which we proposed a simple scaling method to test structural identifiability [2]. Our scaling invariance method (SIM) tests for scaling symmetries only, and Villaverde and Massonis correctly show the SIM may fail to detect identifiability problems when a model has other types of symmetries (we indeed indicated but not investigated the importance of generalizing the method to other symmetries). Thus, we agree that our simple method provides a necessary but not sufficient condition for identifiability, and we appreciate their careful analysis and constructive criticism.We nevertheless think that the simple method remains useful because it is so simple. Even for investigators with little training in mathematics, the method provides a necessary condition for structural identifiability that can be derived in a few minutes with pen and paper. Similarly, we have found its pedagogic strength by teaching the method to our own graduate students and colleagues. More advanced methods (such as STRIKE-GOLDD [3,4], COMBOS [5], or SIAN [6]) are typically intimidating for researchers with a background in Biology or Bioinformatics. This simple method can help those practitioners to familiarize themselves with the identifiability problem and better understand their models.Finally, it is worth noting that if scaling invariance is the only symmetry (as it was in all the cases we analyzed), our SIM remains valuable (albeit uncontrolled), and surprisingly effective for a wide variety of problems (as the extensive list collected in the Supplementary Material our paper [2]). We guess that the SIM especially fails when applied to linear models (as more potential rotations of the variables leave the system invariant), and in non-linear scenarios where some parameters are identical. For instance, the FitzHugh-Nagumo model raised by Villaverde and Massonis, x˙1(t)=c(x1(t)−x13(t)3−x2(t)+d),x˙2(t)=1c(x1(t)+a−b·x2(t)),y(t)=x1(t), could have been written as x˙1(t)=λ1x1(t)−λ2x13(t)3−λ3x2(t)+d,x˙2(t)=λ4x1(t)+a−b·x2(t),y(t)=x1(t) where λ₁ = λ₂ = λ₃ = 1/λ₄ = c. One of the reasons why our method fails, in this case, might be these additional symmetries introduced in this more elaborate notation of the model.Hence, it is worth understanding generic conditions under which the SIM method is expected to be fragile, possibly using STRIKE-GOLDD to test large families of nonlinear models.As a final remark, we appreciate that Villaverde and Massonis have shared their source code, so researchers might have a gold standard to test identifiability.     

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.     

Spatial-Temporal Variation and Primary Ecological Drivers of Anopheles sinensis Human Biting Rates in Malaria Epidemic-Prone Regions of China

Background

Robust malaria vector surveillance is essential for optimally selecting and targeting vector control measures. Sixty-two vector surveillance sites were established between 2005 and 2008 by the national malaria surveillance program in China to measure Anopheles sinensis human biting rates. Using these data to determine the primary ecological drivers of malaria vector human biting rates in malaria epidemic-prone regions of China will allow better targeting of vector control resources in space and time as the country aims to eliminate malaria.

Methods

We analyzed data from 62 malaria surveillance sentinel sites from 2005 to 2008. Linear mixed effects models were used to identify the primary ecological drivers for Anopheles sinensis human biting rates as well as to explore the spatial-temporal variation of relevant factors at surveillance sites throughout China.

Results

Minimum semimonthly temperature (β = 2.99; 95% confidence interval (CI) 2.07- 3.92), enhanced vegetation index (β =1.07; 95% CI 0.11–2.03), and paddy index (the percentage of rice paddy field in the total cultivated land area of each site) (β = 0.86; 95% CI 0.17–1.56) were associated with greater An. Sinensis human biting rates, while increasing distance to the nearest river was associated with lower An. Sinensis human biting rates (β = −1.47; 95% CI −2.88, −0.06). The temporal variation (σt02=1.35) in biting rates was much larger than the spatial variation (σs02=0.83), with 19.3% of temporal variation attributable to differences in minimum temperature and enhanced vegetation index and 16.9% of spatial variance due to distance to the nearest river and the paddy index.

Discussion

Substantial spatial-temporal variation in An. Sinensis human biting rates exists in malaria epidemic-prone regions of China, with minimum temperature and enhanced vegetation index accounting for the greatest proportion of temporal variation and distance to nearest river and paddy index accounting for the greatest proportion of spatial variation amongst observed ecological drivers.

Conclusions

Targeted vector control measures based on these findings can support the ongoing malaria elimination efforts in China more effectively.     

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.     

Anti-Inflammatory and Antinociceptive Properties of Flavonoids from the Fruits of Black Mulberry (Morus nigra L.)

We analyzed the anti-inflammatory and antinociceptive activities of total flavonoids (TF) found in black mulberry fruits. The TF content was 20.9 mg/g (dry weight). Two anthocyanins, cyanidin-3-O-glucoside (8.3 mg/g) and cyanidin-3-O-rutinoside (2.9 mg/g), were identified in the fruits by UPLC. The TF of black mulberry fruits had significant reducing power and radical (OH^-,O2.−, DPPH and ABTS) scavenging activities that was demonstrated in a dose-response curve. The TF had inhibitory activities on xylene-induced ear edema and carrageenan-induced paw edema in mice. In addition, TF had antinociceptive activities in the two nociceptive phases of formalin test. We used ELISA to detect the pro-inflammatory cytokines IL-1β, TNF-α, IFN-γ, and NO in the serum of mice. These cytokines were significantly inhibited or scavenged by TF (50 and 100 mg/kg). The results demonstrated that TF of black mulberry possess anti-inflammatory and analgesic effects that might correlate to its antioxidant activities and inhibition of pro-inflammatory cytokines.     

Influence of Adjuvant Therapy in Cancer Survivors on Endothelial Function and Skeletal Muscle Deoxygenation

The cardiotoxic effects of adjuvant cancer treatments (i.e., chemotherapy and radiation treatment) have been well documented, but the effects on peripheral cardiovascular function are still unclear. We hypothesized that cancer survivors i) would have decreased resting endothelial function; and ii) altered muscle deoxygenation response during moderate intensity cycling exercise compared to cancer-free controls. A total of 8 cancer survivors (~70 months post-treatment) and 9 healthy controls completed a brachial artery FMD test, an index of endothelial-dependent dilation, followed by an incremental exercise test up to the ventilatory threshold (VT) on a cycle ergometer during which pulmonary V˙O2 and changes in near-infrared spectroscopy (NIRS)-derived microvascular tissue oxygenation (TOI), total hemoglobin concentration ([Hb]_total), and muscle deoxygenation ([HHb] ≈ fractional O₂ extraction) were measured. There were no significant differences in age, height, weight, and resting blood pressure between cancer survivors and control participants. Brachial artery FMD was similar between groups (P = 0.98). During exercise at the VT, TOI was similar between groups, but [Hb]_total and [HHb] were significantly decreased in cancer survivors compared to controls (P < 0.01) The rate of change for TOI (ΔTOIΔ/V˙O2) and [HHb] (Δ[HHb]/ΔV˙O2) relative to ΔV˙O2 were decreased in cancer survivors compared to controls (P = 0.02 and P = 0.03 respectively). In cancer survivors, a decreased skeletal muscle microvascular function was observed during moderate intensity cycling exercise. These data suggest that adjuvant cancer therapies have an effect on the integrated relationship between O₂ extraction, V˙O2 and O₂ delivery during exercise.     

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.     

The Impact of Prior Information on Estimates of Disease Transmissibility Using Bayesian Tools

The basic reproductive number (R₀) and the distribution of the serial interval (SI) are often used to quantify transmission during an infectious disease outbreak. In this paper, we present estimates of R₀ and SI from the 2003 SARS outbreak in Hong Kong and Singapore, and the 2009 pandemic influenza A(H1N1) outbreak in South Africa using methods that expand upon an existing Bayesian framework. This expanded framework allows for the incorporation of additional information, such as contact tracing or household data, through prior distributions. The results for the R₀ and the SI from the influenza outbreak in South Africa were similar regardless of the prior information (R^0 = 1.36–1.46, μ^ = 2.0–2.7, μ^ = mean of the SI). The estimates of R₀ and μ for the SARS outbreak ranged from 2.0–4.4 and 7.4–11.3, respectively, and were shown to vary depending on the use of contact tracing data. The impact of the contact tracing data was likely due to the small number of SARS cases relative to the size of the contact tracing sample.     

Ultrashort echo imaging of cyclically loaded rabbit patellar tendon

Tendinopathy affects individuals who perform repetitive joint motion. Magnetic resonance imaging (MRI) is frequently used to qualitatively assess tendon health, but quantitative evaluation of inherent MRI properties of loaded tendon has been limited. This study evaluated the effect of cyclic loading on _T2^?${T_2}^{?}$ values of fresh and frozen rabbit patellar tendons using ultra short echo (UTE) MRI. Eight fresh and 8 frozen rabbit lower extremities had MR scans acquired for tendon _T2^?${T_2}^{?}$ evaluation. The tendons were then manually cyclically loaded for 100 cycles to 45 N at approximately 1 Hz. The MR scanning was repeated to reassess the _T2^?${T_2}^{?}$ values. Analyses were performed to detect differences of tendon _T2^?${T_2}^{?}$ values between fresh and frozen samples prior to and after loading, and to detect changes of tendon _T2^?${T_2}^{?}$ values between the unloaded and loaded configurations. No difference of _T2^?${T_2}^{?}$ was found between the fresh and frozen samples prior to or after loading, p=0.8 and p  =0.1, respectively. The tendons had significantly shorter _T2^?${T_2}^{?}$ values, p  =0.023, and reduced _T2^?${T_2}^{?}$ variability, p  =0.04, after cyclic loading. Histologic evaluation confirmed no induced tendon damage from loading. Shorter _T2^?${T_2}^{?}$, from stronger spin–spin interactions, may be attributed to greater tissue organization from uncrimping of collagen fibrils and lateral contraction of the tendon during loading. Cyclic tensile loading of tissue reduces patellar tendon _T2^?${T_2}^{?}$ values and may provide a quantitative metric to assess tissue organization.