Increased labile iron pool in sorghum embryonic axes after the exogenous application of nitric oxide is independent on the nature of the NO donor

The objective of this work was to explore the hypothesis that nitric oxide (NO) affects Fe bioavailability in sorghum (Sorghum bicolor (L.) Moench) embryonic axes. NO content was assessed in embryonic axes isolated from seeds control or exposed to NO-donors, employing spin trapping electron paramagnetic resonance (EPR) methodology. NO donors such as sodium nitroprusside (SNP) and diethylenetriamine NONOate (DETA NONOate), released NO that permeated inside the axes increasing NO content. Under these conditions low temperature EPR was employed to study the labile iron pool. A 2.5 fold increase was observed in NO steady state concentration after 24 h of exposure to NO donors that was correlated to a 2 fold increase in the Fe labile pool, as compared to control axes. This observation provides experimental evidence for a potential role of NO in Fe homeostasis.Key words: iron, labile iron pool, nitric oxide, sorghumNitric oxide (NO) has a wide range of functions, among them promotion of growth and seed germination were described in several plant species.¹ Evidences for its participation in Fe homeostasis in planta arise from the fact that Fe deficiency can be reverted enhancing NO level.² Moreover, it is expected that NO acts as intercellular messenger³ being transported from the site of its synthesis. Nitrosylated Fe complexes, formed by reaction of NO with Fe²⁺ and biological thiols, have been proposed as NO carriers, since they are relative stable molecules.⁴The ability of Fe of changing its oxidation state and redox potential in response to changes in the nature of the ligand makes this metal essential for almost all living organisms.⁵ Fe-containing enzymes are the key components of many essential biological reactions. However, the same biochemical properties that make Fe beneficial might be a drawback in some particular conditions, when improperly shielded Fe can catalyze one-electron reductions of O₂ species that lead to the production of reactive free radicals. The toxicity of Fe depends on the Fenton reaction, which produces the hydroxyl radical (·OH) or an oxoiron compound (LFeO²⁺) and on its reactions with lipid hydroperoxides.⁶Most of the current information about NO functions in plants comes from pharmacological studies using NO donors, which generate NO either spontaneously, or after metabolic activation. Moreover, NO production from numerous compounds strongly depends on pH, temperature, light and the presence of reductants.⁷ SNP and DETA NONOate have different kinetics and mechanisms of NO release. However, both are suitable compounds for long-term treatments, since their stability is higher than other NO donors.In this work we evaluated NO steady state concentration in sorghum embryonic axes 24 h after imbibition, in control seeds (distilled water) and in seeds placed either in 1 mM SNP or DETA NONOate. SNP contains Fe in its chemical structure, thus a control was carried out employing photodegraded SNP, which consist of 1 mM SNP solution which had been left under light until all NO was released from the molecule. As it is shown in

Quantitative Analysis of Three-Dimensional Fluorescence Localization Microscopy Data

Identifying the three-dimensional molecular organization of subcellular organelles in intact cells has been challenging to date. Here we present an analysis approach for three-dimensional localization microscopy that can not only identify subcellular objects below the diffraction limit but also quantify their shape and volume. This approach is particularly useful to map the topography of the plasma membrane and measure protein distribution within an undulating membrane.Single molecule localization microscopy (SMLM) (1–3) is a superresolution fluorescence microscopy technique that produces coordinate data for single molecule localizations with a precision of tens of nanometers in live and fixed cells. These methods have mainly been performed with total internal reflectance fluorescence microscopy and therefore have generated two-dimensional molecular coordinates. Such two-dimensional data sets have revealed nanosized clusters of membrane proteins at the cell surface (4–7). This was achieved with analysis routines based on pair-correlation analysis (8), Ripley’s K function (9), and related techniques. While three-dimensional localization microscopy techniques such as biplane imaging (10), astigmatic spot analysis (11), and depth-encoding point-spread functions (12) have now been developed, quantitative analysis approaches of three-dimensional coordinate patterns have not.Here, we describe an approach based on Getis and Franklin''s local point pattern analysis to quantitatively analyze three-dimensional subcellular structures and map plasma membrane topography. The latter can also be used to account for topography-induced clustering of membrane proteins in an undulating membrane. To illustrate the approach, we generated three-dimensional SMLM data of the membrane dye DiI and the protein Linker for Activation of T cells (LAT) fused to the photoswitchable fluorescent protein mEos2 in T cells. It has been previously shown that LAT resides within the plasma membrane as well as membrane-proximal vesicles (5,13). The data were acquired using the biplane SMLM technique and highly inclined and laminated optical sheet illumination (14). Three-dimensional molecular coordinates were calculated by fitting a three-dimensional theoretical point-spread-function to the acquired data.As previously described for two-dimensional SMLM data analysis (5), Ripley’s K-function is calculated according to Eq. 1 where V is the analyzed volume, n is the total number of points, and r is the radius of a sphere (a circle for the two-dimensional case) centered on each point. The value K(r) is thus a measure of how many points are encircled within a sphere of radius r:K(r)=V∑i=1n∑j=1n(δij/n2);δij=1ifd(pointi,pointj)<r,0else.(1)For completely spatially random (CSR) data, K(r) scales with the volume of the sphere. We therefore linearize the K-function such that it scales with radius (the L-function) using:L(r)=(3K(r)4π)1/3.(2)The value of L(r)−r is then zero for the CSR case. Values of L(r)−r above zero indicate clustering at the length scale, r.Next we used the related Getis and Franklin''s local point pattern analysis to generate a clustering value (L(r) at r = 50 nm; L(50)) for each point, j, based on the local three-dimensional molecular density. This was calculated using:Lj(50)=((3V4π)∑i=1n(δijn))1/3;δij=1ifd(pointi,pointj)<50,0else.(3)These values can then be interpolated such that every voxel in a volume is assigned a cluster value based on the number of encircled points, relative to the expected CSR case. This allows construction of isosurfaces where all points on the surface have an identical L(50) value. A high threshold imparts a strict criterion for cluster detection compared to a lower one, and this allows users to, for example, determine the efficiency of sequestration into clusters by quantifying the cluster number and size as a function of the threshold.To illustrate the identification of subcellular structures, Lat-mEos2 was imaged by three-dimensional SMLM in activated T cells at the immunological synapse (Fig. 1 A). Three-dimensional projections of isosurfaces (for L(50) = 200) clearly identified intracellular LAT vesicles at varying depths within the synapse (Fig. 1, B and C). Cluster statistics were extracted from this data set to quantify the distribution of clusters in the z direction as well as the volume and sphericity of the LAT objects themselves (Fig. 1, D–F).Open in a separate windowFigure 1Identification of subcellular objects in three dimensions by isosurface rendering of molecular distribution. (A) Schematic of a T cell synapse formed against an activating coverslip where subsynaptic LAT vesicles (red dots) can be imaged with three-dimensional SMLM. (B and C) Isosurfaces, shown in x,z view (B) and as projection (C), identify T cell vesicles as LAT objects with L(50) > 200 (Eq. 3). (D–F) Distribution of LAT objects in z direction (D), volume (E), and sphericity (F) of LAT objects in T cells.Membrane undulations can cause clustering artifacts when the distribution of membrane proteins is recorded as a two-dimensional projection (15) (Fig. 2 A), as is the case in two-dimensional SMLM under total internal reflectance fluorescence illumination. To illustrate a solution to this problem, we obtained three-dimensional SMLM data sets of the membrane dye DiI (16) in resting T cells adhered onto nonactivating coverslips. With appropriately short labeling times to prevent dye internalization, it can be assumed that all DiI molecules reside in the plasma membrane. In this case, as is the case for plasma membrane proteins, neither two-dimensional nor three-dimensional analysis is appropriate, as it is a priori known the points must be derived from a two-dimensional membrane folded in three-dimensional space. To correct for membrane undulations, the plasma membrane topography must first be mapped so that molecular coordinates of membrane molecules can be appropriately corrected in two-dimensional projections. The position of the plasma membrane in three dimensions, i.e., the membrane topography, was determined by averaging the z position of all DiI molecules within a 100-nm radius in x-y at each point. The averaged z-position of DiI molecules was then displayed as a map, which exhibits a smooth, undulating profile (Fig. 2 B). The selection of this radius determines the accuracy of the assigned z position but also causes smoothing of the membrane profile.Open in a separate windowFigure 2Mapping of membrane topography and correction of molecular distributions in undulating membranes. (A) Two-dimensional projections can cause cluster artifacts, for example in membrane ruffles. Molecules (red rectangles) in the upper image are equally spaced along the membrane but appear as clusters in two-dimensional projections in areas with high gradient. (B) Three-dimensional membrane topography of a 2 × 2 μm plasma membrane area of a resting T cell obtained from averaged z positions of DiI molecules. Note that membrane undulation is ∼100 nm. (C) Map of membrane gradient, corresponding to the topography map shown in panel B, with an area of high gradient highlighted (dashed red box). (D) Correction of the circle radii in the Getis and Franklin cluster map calculations to account for projection artifacts. (E and F) Cluster map of data shown in panel C before (E) and after (F) correction for membrane gradient. Boxes in panels C, E, and F highlight the regions with high membrane gradient.Next, the gradient at the position of each DiI molecule was determined and interpolated into a gradient map (Fig. 2 C). Here, blue represents horizontal, i.e., flat membrane areas, whereas red regions indicate areas of high gradient. The information from the gradient map was then used to ensure that the two-dimensional circles in the Getis and Franklin cluster map calculations each correspond to an identical area of membrane, hence accounting for two-dimensional projection artifacts. To do this, the size of the circle (r) used to calculate the L value for each molecule was modified using Eq. 4, where c is calculated for the surface, S, using Eq. 5:r(corr)=r(uncorr)(1+c2)1/4,(4)c=((∂S∂x)2+(∂S∂y)2)1/2.(5)This operation is shown schematically in Fig. 2. The comparison of Getis and Franklin cluster maps before (Fig. 2 E) and after (Fig. 2 F) correction for the gradient shows that cluster values for DiI molecules were substantially reduced by up to 5–10% at sites where the plasma membrane had a high gradient (area highlighted in red box), and where the two-dimensional projection of three-dimensional structures caused an overestimation of clustering.In conclusion, we demonstrated that three-dimensional superresolution localization microscopy data can be used to identify and quantify subcellular structures. The approach has the distinct advantage that subcellular structures are solely identified by the distribution of the fluorescent marker so that no a priori knowledge of the structure is necessary. How precisely the subcellular structures are identified only depends on how efficiently the fluorescent maker is recruited to the structure, and hence does not depend on the resolution limits of optical microscopy. We applied the methods to two very different structures in T cells: small intracellular vesicles and the undulating plasma membrane. Importantly, the topography of plasma membrane can also be used to correct clustering artifacts in two-dimensional projections, which may be useful for distribution analysis within membranes.     

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.     

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.     

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.     

On the Adjacent Eccentric Distance Sum Index of Graphs

For a given graph G, ε(v) and deg(v) denote the eccentricity and the degree of the vertex v in G, respectively. The adjacent eccentric distance sum index of a graph G is defined as ξsv(G)=∑v∈V(G)ε(v)D(v)deg(v), where D(v)=∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. In this paper we derive some bounds for the adjacent eccentric distance sum index in terms of some graph parameters, such as independence number, covering number, vertex connectivity, chromatic number, diameter and some other graph topological indices.     

Reactive Nitrogen Species-Dependent Effects on Soybean Chloroplasts

Nitric oxide (NO) generation by soybean (Glycine max, var ADM 4800) chloroplasts was studied by electron paramagnetic resonance (EPR) spin-trapping technique.¹ Both nitrite and L-arginine (arg) are the required substrates for enzymatic activities considered as possible sources of NO in plants. Soybean chloroplasts showed a NO production of 3.2 ± 0.2 nmol min⁻¹ mg⁻¹ protein in the presence of 1 mM NaNO₂. Chloroplasts incubated with 1 mM arg showed a NO production of 0.76 ± 0.04 nmol min⁻¹ mg⁻¹ protein. This production was inhibited when chloroplasts were incubated in presence of NOS-inhibitors L-NAME and L-NNA. In vitro exposure of chloroplasts to a NO-donor (GSNO) decreased both ascorbyl radical content and the activity of ascorbate peroxidase, without modification of the total ascorbate content. Exposure of the isolated chloroplasts to a NO-donor decreased lipid radical content in membranes, however, incubation in the presence of 25 µM peroxynitrite (ONOO⁻) led to an increase in lipid-derived radicals (34%). The effect of ONOO⁻ on protein oxidation was determined by western blotting, showing an increase in carbonyl content either in stroma or thylakoid proteins as compared to control. Taken as a whole, NO seems to be an endogenous metabolite in soybean chloroplasts and reactive nitrogen species could exert either antioxidant or prooxidant effects on chloroplasts, since both a decreased lipid radical content in membranes and a decrease in the activity of ascorbate peroxidase were observed after exposure to a NO donor.Key Words: ascorbate, ascorbate peroxidase, chloroplasts, nitric oxide, peroxynitriteThe origin of nitric oxide (NO) in plants under aerobic conditions is currently under study. Although plants with low level of arginine shows arg-stimulated NO accumulation,² the mechanism for arginine-dependent NO synthesis in plants is still unknown, because the detection of an animal-type NOS remains elusive to date.^3,4 Even though assimilatory nitrate reductase is an enzymatic source of NO, its role in vivo would be limited by both its cytosolic localization which difficult the availability for nitrite, and the relative high K_m for nitrite (100 µM).⁵Chloroplasts have been previously marked as NO sources based in nonquantitative studies employing fluorescence microscopy^6,7 and immunogold electron microscopy.⁸ In our work we employed an specific technique (EPR, electron paramagnetic resonance with spin trap⁹) to detect NO as an endogenous metabolite and to quantify its generation in the presence of different substrates. In order to gain insight on the mechanism leading to NO production both nitrite-dependent and arg-dependent pathways were evaluated. In the presence of 1 mM arg and 0.1 mM NADPH the rate of NO generation was 0.76 ± 0.04 nmol min⁻¹ mg⁻¹ prot (arg-dependent synthesis). The synthesis of NO resulted completely blocked in the presence of arg analogs (L-NAME and L-NNA). It is important to point out that the content of arg in the chloroplasts stroma is high as compared to the content of other amino acids (56.7 ± 0.8 nmol mg⁻¹ prot), suggesting that this pathway could be operative under physiological conditions.Soybean chloroplasts showed a NO production of 3.2 ± 0.2 nmol min⁻¹ mg⁻¹ prot in the presence of 1 mM NaNO₂. Furthermore, NO generation was detected in the presence of nitrite concentrations as low as 25 mM. Since nitrite-dependent NO generation resulted inhibited by 50% by the addition of DCMU, and no NO generation was measured in the stroma fraction, thylakoidal electron transport seems to be a key feature in NO synthesis.According to this scenario and assuming that the two independent pathways for NO generation in chloroplasts are operative, the total rate of production of NO could be understood as the generation by the activity of an arg-dependent enzyme and by a NO₂⁻ dependent pathway, as indicated by eq. 1.d[NO]dt=(d[NO]dt)NOS like+(d[NO]dt)NO2−(1)Regarding the NO disappearance, from a kinetic point of view, the rate of the reaction of NO with O₂⁻ to generate peroxynitrite seems to be the main pathway, since the reaction is diffusionally controlled. Thus, the rate of disappearance of NO could be estimated from the rate of generation of ONOO⁻ (eq. 2); however, other reactions should be considered under nonphysiological conditions.−d[NO]dt=d⌊ONOO−⌋dt=k[NO][O2−](2)NO generation rate should be equal to NO consumption rate in order to keep a physiological NO steady state concentration (eq. 3)d[NO]dt=−d[NO]dt(3)Thus, replacing NO generation and disappearance rates by those rates indicated in equations 1 and 2, (d[NO]dt)NOS like+(d[NO]dt)NO2−=k[NO][O2−](4) The data obtained under unrestricted availability of substrates, indicate a generation rate of NO by the activity of a NOS-like enzyme of 13 × 10⁻⁹ M s⁻¹. Chloroplastic NO generation rate in the presence of 100 µM NO₂⁻ was 14 × 10⁻⁹ M s⁻¹. Thus, according to equation 1, the rate of generation of NO is approximately 3 × 10⁻⁸ M s⁻¹. Assuming a steady state concentration for O₂⁻ of 1 nM in chloroplasts¹⁰ and a rate constant (k) of 6.9 × 10⁹ M⁻¹ s⁻¹ for the reaction between O₂⁻ and NO,¹¹ a steady state concentration of 4 nM for NO in the chloroplast could be estimated. Since under in vivo conditions chloroplasts may content the required substrates for the NO synthesis, the assays presented here strongly suggest that a feasible NO production could take place inside the chloroplasts. However, nonsupplemented chloroplasts did not show any NO-dependent EPR signal. This observation agrees with the fact that NO steady state concentration under physiological conditions as was calculated here (4 nM) is below the EPR detection limit (500 nM).¹²Further studies should be performed to characterize NO oxidative effects on chloroplasts. Scavenging of O₂⁻ and H₂O₂ is essential for chloroplasts to maintain their ability to fix CO₂ since several enzymes in the CO₂-reduction cycle are sensitive to active oxygen species.¹³ These organelles lacking catalase, contain a significant peroxidase activity.¹⁴ H₂O₂-reduction catalized by ascorbate peroxidase (AP) lead to ascorbate oxidation and produces ascorbyl radical (A^.).¹⁵ In isolated chloroplast the content of A^.. was evaluated in DMSO based extract by EPR¹⁶. Quantification of EPR signals indicated that A^. content in control chloroplasts (123 ± 5 pmol mg⁻¹ prot) decreased after exposure to NO (Fig. 1). The total content of ascorbate, assessed by an HPLC technique¹⁷ in chloroplasts isolated from soybean leaves exposed to NO was not significantly different from the measured content in chloroplasts not exposed to the NO donor (Fig. 1). The activity of AP was significantly decreased by 48, 53 and 54% after exposure of the chloroplasts to NO-donor. Previous data suggested that AP could be inactivated by NO via oxidation of functional thiols.¹⁸ Besides, the reversible inhibition of AP could be due to the formation of Fe-nitrosyl complexes between NO and the Fe atom of the heme group, as it was previously described for NO-mediated activation of guanylate cyclase and the inhibition of cytochrome P₄₅₀ and catalase in mammals.¹⁹ The data presented here showed that in isolated chloroplasts exposed to a NO donor, there could be either a limited damage associated to the decrease in the content of A^.. or an increased cellular deterioration by the decrease in the activity of the enzyme responsible for the scavenging of H₂O₂.Open in a separate windowFigure 1Ascorbate metabolism in soybean chloroplasts after NO exposure. A^.. content (▪), ascorbate content (▪), and AP activity (*) as a function of the exposure of isolated chloroplasts to GSNO in the presence of 50 µM DTT. * = significantly different at p ≥ 0.05 from the value obtained in the absence of GSNO + 50 µM DTT.Thus, in situ generation of NO could play a protective role in preventing the oxidation of chloroplastic lipids; however, the reaction of NO with O₂⁻ leading to ONOO⁻ production may result in a potential source of damage or as it is shown here by the significant decrease of the AP activity that consumes H₂O₂. NO is a suitable candidate to modulate cellular H₂O₂ level through the chloroplast function, as an initial step to regulate complex metabolic pathways directed to activate physiological responses, defense pathways or deleterious effects in the cytosol. Furthermore, an integrated study on the effect of nitrogen reactive species is required under stress conditions to characterize the metabolic pathways involved in the resulting cellular damage.     

Introducing Titratable Water to All-Atom Molecular Dynamics at Constant?pH

Recent development of titratable coions has paved the way for realizing all-atom molecular dynamics at constant pH. To further improve physical realism, here we describe a technique in which proton titration of the solute is directly coupled to the interconversion between water and hydroxide or hydronium. We test the new method in replica-exchange continuous constant pH molecular dynamics simulations of three proteins, HP36, BBL, and HEWL. The calculated pK_a values based on 10-ns sampling per replica have the average absolute and root-mean-square errors of 0.7 and 0.9 pH units, respectively. Introducing titratable water in molecular dynamics offers a means to model proton exchange between solute and solvent, thus opening a door to gaining new insights into the intricate details of biological phenomena involving proton translocation.Solution pH is an important factor in biology. Although neutral pH in extracellular medium accounts for balanced electrostatics and proper folding of protein structures, pH gradients across cell membranes induce large conformational changes that are necessary for biological functions, such as ATP synthesis and efflux of small molecules out of the cell. To gain detailed insights into pH-dependent conformational phenomena, several constant pH molecular dynamics (pHMD) methods, based on either discrete or continuous titration coordinates, have been developed in the last decade (1–4). In the continuous pHMD (CpHMD) framework (2,4), a set of titration coordinates {λ_i} are simultaneously propagated along with the conformational degrees of freedom. Although the original CpHMD method based on the generalized Born (GB) implicit-solvent models (2,4) offers quantitative prediction of pK_a values and pH dependence of folding and conformational dynamics of proteins (5), its accuracy and applicability to highly charged systems and those with dominantly hydrophobic regions are limited due to the approximate nature of the underlying implicit-solvent models.Motivated by the above-mentioned need, three groups have made efforts to develop a CpHMD method using exclusively the explicit-solvent models (6–8). In our development, the titration of acidic and basic sites is coupled with that of coions to level the total charge of the system (8). To further improve physical realism, here we replace the coions by titratable water molecules, which not only absorb the excess charge but also enable direct modeling of solute-solvent proton exchange in classical molecular dynamics simulations.To illustrate the utility of the new methodology, we applied it to the titration simulations of three proteins that were previously used to benchmark the GB-based CpHMD. Although this work does not explore specific interactions between titratable waters and proteins, the methodology can be further tested or improved to provide a rigorous way for modeling proton transfer in molecular dynamics, which is a computationally efficient alternative to the empirical valence-bond theory-based methodologies (9,10).We define titration of water as:

• 1.Loss of a proton to give a negatively charged hydroxide,

H₂O ? OH^? + H⁺, (1)or

• 2.Gain of a proton to give a positively charged hydronium,

H₂O + H⁺ ? H₃O⁺.(2)We now couple the titration of hydroxide (Eq. 1) with that of an acidic site of the solute in the CpHMD simulation,HA+OH−⇌KaA−+H2O.(3)The use of hydronium is avoided here to prevent a potential artifact due to prolonged attraction with A⁻. Analogously, we couple the titration of hydronium (Eq. 2) with that of a basic site,BH++H2O⇌KbH3O++B.(4)Thus, effectively, a proton is transferred between the solute and solvent. However, we should note that in CpHMD simulations, titratable protons are represented by covalently attached dummies (2,4). Through varying the atomic charges and van der Waals interactions, they are seen by other atoms in the protonated state but not in the unprotonated state (see Table S1 in the Supporting Material). Furthermore, the solution proton concentration is implicitly modeled through a free energy term (2,4).In CpHMD, the reference potential of mean force (PMF) for titration is that of the model compound (blocked single amino acid in water) along λ (2,4). In the presence of cotitrating water molecules, it is necessary to add the PMF for the conversion of water to hydroxide or hydronium. One-nanosecond NPT simulations at ambient pressure and temperature were performed to calculate the average force, 〈dU/d,θ〉 at given θ-values, which are related to λ by λ = sin² θ (see Fig. S1 in the Supporting Material). Thermodynamic integration was then applied to calculate the PMF. We found that the average force can be accurately fit when assuming the PMF is quadratic in λ (Fig. 1). The same applies to the PMFs for titration of models Asp, Glu, and His. After testing on the titration of model compounds (see Table S2), we performed 10-ns all-atom CpHMD simulations with the pH replica-exchange protocol for three proteins: HP36, BBL and HEWL (see the Supporting Material for details). Most of the calculated pK_a values were converged in 10 ns per replica (see Fig. S3). Results are summarized in Fig. S4. Based on the 10-ns data, the root-mean-square (RMS) and average absolute errors are 0.9 and 0.7 pH units, respectively, while the largest absolute error is 2.5 (Glu³⁵ of HEWL). Linear regression of the calculation versus experiment gives R² of 0.8 and slope of 1.2.Open in a separate windowFigure 1Average force and potential of mean force for converting a water molecule to hydroxide (A) and hydronium. (B) (Data points) Average forces. (Dashed curves) Best fits using a linear function, 2A(λ − B). (Solid curves) Corresponding potential of mean force.

Table 1

Calculated and experimental pK_a values of three proteins

The Physical Foundation of Vasoocclusion in Sickle Cell Disease

The pathology of sickle cell disease arises from the occlusion of small blood vessels because of polymerization of the sickle hemoglobin within the red cells. We present measurements using a microfluidic method we have developed to determine the pressure required to eject individual red cells from a capillary-sized channel after the cell has sickled. We find that the maximum pressure is only ∼100 Pa, much smaller than typically found in the microcirculation. This explains why experiments using animal models have not observed occlusion beginning in capillaries. The magnitude of the pressure and its dependence on intracellular concentration are both well described as consequences of sickle hemoglobin polymerization acting as a Brownian ratchet. Given the recently determined stiffness of sickle hemoglobin gels, the observed obstruction seen in sickle cell disease as mediated by adherent cells can now be rationalized, and surprisingly suggests a window of maximum vulnerability during circulation of sickle cells.Human capillaries are narrower than the erythrocytes they convey. In sickle cell disease, red cells can become rigid in those capillaries, because the hemoglobin inside the red cell will aggregate into stiff polymers. This happens once the molecules deliver their oxygen, and led to the long-held view that capillary occlusion was central to the pathophysiology of the disease (1,2). This was challenged when microscopic study of animal model tissues perfused with sickle blood revealed blockages that began further downstream, in the somewhat larger venules (3–5), at the site of adherent red or white cells which diminished the vessel lumen without fully obstructing the flow. Yet no rationale has been presented for the failure of the prior assumption of capillary blockage. Microfluidic methods (6) are ideally suited to discover why cells don’t get stuck in the capillaries, yet occlude subsequent vessels, and we have constructed a system to address this question. Our measurements show that the pressure differences across capillaries in vivo can easily dislodge a cell sickled within a capillary, giving an experimental answer to the question of why sickled cells don’t stick in capillaries. It turns out that the pressure a cell can withstand is quantitatively explained by the Brownian ratchet behavior of sickle hemoglobin polymerization.We constructed single-cell channels in transparent polydimethylsiloxane, with a cross section (1.5 μm × 4 μm) that is smaller than the resting diameter of red cells (Fig. 1). These channels are much narrower than those that have been employed in other recent studies of the sickling process (7,8), and they resemble human capillaries in permitting only one cell at a time to pass through them. We used a laser photolysis method to create ligand free (deoxygenated) cells, and this requires that the hemoglobin bind CO, which can then be readily removed by strong illumination, in contrast to bound O₂ which is released with far lower efficiency than CO. The microfluidic chips were enclosed in a gas-tight chamber flushed with CO to avoid introduction of oxygen and keep the cells fully ligated before photolysis. The profiles of the channels were confirmed by microscopic observation. To confirm that liquid did not pass around the cells when they were trapped in the channels, fluorescent beads were introduced into some cell solutions. The beads did not pass the cells, nor did they approach the cell when it was occluded, verifying that no significant flow occurred around the cell when it was stuck.Open in a separate windowFigure 1An erythrocyte enters a channel (moving left to right) and is positioned in the center, where it will be photolyzed. The channel cross section is 1.5 μm × 4 μm, smaller than a resting red cell diameter.Optical measurements were carried out on a microspectrophotometer constructed on an optical table. The system employed ×32 LWD objectives (Leitz, Wetzlar, Germany), which were autofocused during collection of absorption spectra to minimize aberrations. Spectra were obtained using a series 300 camera (Photometrics, Tucson, AZ); video imaging was done with a high-speed camera (Photron, San Diego, CA). Photolysis was provided by a 2020 Argon Ion laser (Spectra Physics, Houston, TX). Sickle cells were obtained from patients at the Marian Anderson Sickle Cell Center at St. Christopher''s Hospital for Children, Philadelphia, PA by phlebotomy into EDTA-containing tubes. The blood was centrifuged at 5°C at 1200g for 4 min, and then the pellet was washed 4× with 15 volumes of buffer (120 mM NaCl, 2 mM KCl, 10 mM dibasic Na Phosphate, 7 mM monobasic Na Phosphate, 3.4 mM Na Bicarbonate, and 6 mM Dextrose) by repeated suspension and centrifugation at 30g for 4 min. This minimizes fibrinogen and platelets in the final suspension, to insure that these studies are controlled by the mechanical properties of the cells themselves.Our experiment began by parking a cell in the center of a channel (Fig. 1). The cell, its hemoglobin, and the microchannel environment all were saturated with CO. Because the thickness of the channel is known, we were able to determine the hemoglobin concentration inside the cell from its absorption spectrum (Fig. 2 A). Steady-state laser illumination then removed the CO, allowing the hemoglobin to polymerize, in which condition it remained while the laser was kept on. Removal of CO was confirmed by observing the spectral difference between COHb and deoxyHb. Photolysis of COHb generates negligible heating (9–11). During illumination, hydrostatic pressure was applied until the cell broke free.Open in a separate windowFigure 2(A) Absorption of the cell (points), fit to a standard spectrum (9). (B) Pressure to dislodge a cell sickled in the microchannel, as a function of intracellular concentration. Note that typical intracellular concentrations are ∼32 g/dL. (Line) Brownian-ratchet theory described in the text. The coefficient of friction (0.036) is within the observed range, and is the only parameter varied.The magnitude of the dislodging pressure, measured by simple height difference between input and output cell reservoirs, is shown in Fig. 2 B. The pressure needed to dislodge the cell increased with increasing intracellular Hb concentration, implying that an increased mass of polymerized hemoglobin is more difficult to dislodge. A clear concentration threshold for capture is apparent. While there is a well-known solubility below which polymers cannot form (18.5 g/dL for the 22°C of this experiment (12)), the threshold here is significantly higher.Central to explaining these observations is a Brownian ratchet mechanism (13) which derives from the metastable nature of this polymerization process. Unless disrupted, as by centrifugation, polymerization in sickle hemoglobin terminates before the thermodynamic limit of monomer solubility is reached (14,15). This arises from the fact that polymers only grow at their ends, which are easily occluded in the dense mass of polymers that form. This end obstruction leaves the system in a metastable state and fluctuations accordingly provide polymers with space into which they can incrementally grow. This Brownian ratchet has been shown to lead to dramatic fiber buckling when individual fibers are isolated in sickle cells (16). The force can be simply expressed as f = (kT/δ) ln S(c), where k is Boltzmann’s constant, T the absolute temperature, δ the net spatial elongation from addition of a single monomer, and S is the supersaturation of the solution when the metastable limit is reached, at monomer concentration c. In this calculation, c is taken as the terminal concentration, computed from our empirical finding (15) that in this metastable system the amount of polymerized hemoglobin Δ is Δ(∞) = 2/3 (c_o-c_s), rather than the expected thermodynamic limit c_o-c_s, where c_o is the initial concentration and c_s is the solubility.For determining the net force, the total number of fibers must be known, and can be calculated based on the double nucleation mechanism (17) which has been quantitatively successful in describing polymerization. The concentration of polymers [p(t)] initially grows exponentially, described by[p(t)]=(AB2J)exp(Bt),where A and B are parameters related to nucleation, and J is the polymer elongation rate, as described in Ferrone et al. (17). Because A and B are both extremely concentration-dependent (9), they will drop dramatically once monomers begin to add to polymers in any significant numbers, and thereby diminish the remaining monomer pool. Thanks to the extreme concentration dependence of the reaction, this rapidly shuts off further polymerization. This happens at approximately the 10th time (the time when the reaction has reached 1/10 of its maximum). Thus, the [p(t_1/10)] ≈ [p(∞)]. Moreover, at one-tenth of the reaction,Δ(t1/10)=12Aexp(Bt1/10)=Δ(∞)10,and thus[p(∞)]=(BJ)(Δ(∞)10)=(BJ)((co−cs)15).For computing the number of fibers, the volume of the cell was taken as 90 μm³. This calculation shows, as expected, that the number of polymers in the cell is highly concentration-dependent, and very few fibers are produced at concentrations just above solubility, but the number grows sharply as concentration rises. This is the main contribution to the threshold in holding force shown by the data.With the force per fiber, and the total number of fibers, the net force against the wall is known. With a coefficient of friction, this reveals the force that a trapped cell can withstand. If the force is divided by the cross-sectional area across which the force is applied, we get a prediction of the dislodging pressure, which can be compared to the data. For a quantitative comparison with the results, two further corrections, of order unity, were applied. Because only normal force will contribute to friction, the calculated force was determined by integrating cos θ. This integration is not over all angles (π) because of the possibility that large incidence angles of the fibers against the wall will lead to fiber runaway (18). Therefore, the integration described is taken to the runaway threshold, here ∼1 rad. Finally, it is necessary to assign a coefficient of friction. Known values span the range of 0.03–0.06 (19). We therefore selected a value within the range, 0.036, as the best match for the data. The predicted pressures match the measurements well, as the line in Fig. 2 B shows.Because the flow resistance is comparable for red cells traversing glass channels and endothelial-lined capillaries (20), we conclude that in vivo the pressures a sickled cell inside a capillary can withstand are no more than hundreds of Pa. This is significantly smaller than typical arteriovenous pressure differentials that have been measured, which range from 0.7 kPa (in hamster skin (21)) to 7.9 kPa (in rat mesentery (22)).Our measurements coupled with recent determination of the stiffness of sickle hemoglobin gels (23) provide the missing physical basis for the processes of vasoocclusion seen in ex vivo tissue and animal models of sickle cell disease, arguing that these observations indeed represent fundamental behavior of sickle cell disease. We now understand this behavior in terms of three possible outcomes, all intimately connected with kinetics:

• 1.Certain escape: A cell that does not polymerize until after passing the obstruction can reach the lungs where it reoxygenates and resets its polymerization clock.
• 2.Possible escape: A cell that polymerizes within the capillary will assume an elongated sausage shape. The forces that it can exert against the wall cannot hold it there, and it will emerge into the postcapillary venule. There it has some chance of passing a subsequent obstruction, though it might also obstruct flow were it to rotate before reaching the adherent cell, so as to present its long dimension to the reduced space it must traverse.
• 3.Certain occlusion: A cell that does not polymerize in the capillary reassumes a larger diameter as soon as it escapes. If the cell then polymerizes before it encounters a cell attached to the venule wall, this rigidified cell will not be able to squeeze past the adherent cell, because that kind of deformation takes MPa (23). This would precipitate the type of blockage that is observed. This suggests that there is a window of greatest vulnerability, toward which therapies might be addressed.