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.     

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.     

An Improved Fst Estimator

The fixation index F _st plays a central role in ecological and evolutionary genetic studies. The estimators of Wright (F^st1), Weir and Cockerham (F^st2), and Hudson et al. (F^st3) are widely used to measure genetic differences among different populations, but all have limitations. We propose a minimum variance estimator F^stm using F^st1 and F^st2. We tested F^stm in simulations and applied it to 120 unrelated East African individuals from Ethiopia and 11 subpopulations in HapMap 3 with 464,642 SNPs. Our simulation study showed that F^stm has smaller bias than F^st2 for small sample sizes and smaller bias than F^st1 for large sample sizes. Also, F^stm has smaller variance than F^st2 for small F _st values and smaller variance than F^st1 for large F _st values. We demonstrated that approximately 30 subpopulations and 30 individuals per subpopulation are required in order to accurately estimate F _st.     

Kinetics of H2O2-driven catalysis by a lytic polysaccharide monooxygenase from the fungus Trichoderma reesei

Owing to their ability to break glycosidic bonds in recalcitrant crystalline polysaccharides such as cellulose, the catalysis effected by lytic polysaccharide monooxygenases (LPMOs) is of major interest. Kinetics of these reductant-dependent, monocopper enzymes is complicated by the insoluble nature of the cellulose substrate and parallel, enzyme-dependent, and enzyme-independent side reactions between the reductant and oxygen-containing cosubstrates. Here, we provide kinetic characterization of cellulose peroxygenase (oxidative cleavage of glycosidic bonds in cellulose) and reductant peroxidase (oxidation of the reductant) activities of the LPMO TrAA9A of the cellulose-degrading model fungus Trichoderma reesei. The catalytic efficiency (kcat/Km(H2O2)) of the cellulose peroxygenase reaction (k_cat = 8.5 s⁻¹, and Km(H2O2)=30μM) was an order of magnitude higher than that of the reductant (ascorbic acid) peroxidase reaction. The turnover of H₂O₂ in the ascorbic acid peroxidase reaction followed the ping-pong mechanism and led to irreversible inactivation of the enzyme with a probability of 0.0072. Using theoretical analysis, we suggest a relationship between the half-life of LPMO, the values of kinetic parameters, and the concentrations of the reactants.     

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

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.     

Epidemic dynamics on semi-directed complex networks

In this paper an SIS model for epidemic spreading on semi-directed networks is established, which can be used to examine and compare the impact of undirected and directed contacts on disease spread. The model is analyzed for the case of uncorrelated semi-directed networks, and the basic reproduction number _R0$R_0$ is obtained analytically. We verify that the _R0$R_0$ contains the outbreak threshold on undirected networks and directed networks as special cases. It is proved that if _R0<1$R_0<1$ then the disease-free equilibrium is globally asymptotically stable, otherwise the disease-free equilibrium is unstable and the unique endemic equilibrium exists, which is globally asymptotically stable. Finally the numerical simulations holds for these analytical results are given.     

The consequences of dominance and gene flow for local adaptation and differentiation at two linked loci

For a subdivided population the consequences of dominance and gene flow for the maintenance of multilocus polymorphism, local adaptation, and differentiation are investigated. The dispersing population inhabits two demes in which selection acts in opposite direction. Fitness is determined additively by two linked diallelic loci with arbitrary intermediate dominance (no over- or underdominance). For weak as well as strong migration, the equilibrium structure is derived. As a special case, a continuous-time continent–island model (CI model) is analyzed, with one-way migration from the continent to the island. For this CI model, the equilibrium and stability configuration is obtained explicitly for weak migration, for strong migration, for independent loci, and for complete linkage. For independent loci, the possible bifurcation patterns are derived as functions of the migration rate. These patterns depend strongly on the degree of dominance. The effects of dominance, linkage, and migration on the amount of linkage disequilibrium (LD) and the degree of local adaptation are explored. Explicit formulas are obtained for $D$   $(=x_1{x}_4-x_2{x}_3)$ and $r^2$ (the squared correlation in allelic state). They demonstrate that dominant island alleles increase $D$ and decrease $r^2$. Local adaptation is elevated by dominance of the locally adaptive alleles. The effective migration rate at a linked neutral locus is calculated. If advantageous alleles are dominant, it is decreased only slightly below the actual migration rate. For a quantitative trait that is determined by two additive loci, the influence of dominance on measures of differentiation is studied. Explicit expressions for $Q_{ST}$ and two types of $F_{ST}$ at equilibrium are deduced and their relation is discussed.     

Analysis of Amylose-borate Interaction by Frontal Gel Filtration

Amylose-borate interaction has been analyzed by frontal gel chromatography, using the constituent velocity data alone. The constituent Velocity equation was reformulated in terms of elution volume for a type of interacting system described by$\text{A}+i\text{B}={\text{AB}}_i(i=1,2,3\dots n)$

Detailed examination of the binding data indicates that, in the complex formation between amylose and borate, this type of equilibria operates predominantly, if not solely. Use of the constituent elution volume equation enabled us, for the first time, to evaluate the association constant (K) and number of binding site pertaining to this system, i.e., K = 4.9 10² and n = 1. There was no evidence indicating the occurrence of the formation of inclusion complex.