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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 equationwhere 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 , yielding(1)Here 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 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) + Gpull(x,t) along the molecular extension x. Here G0(x) is the intrinsic molecular potential and Gpull(x,t) is the potential of the pulling device. When G(x,t) features a high barrier on the scale of kBT (kB 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 functionwhich 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 κS is Gpull(x,t) = κS/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),where k0 is the intrinsic on-rate, β ≡ (kBT)−1, andis the stiffness of the unfolded biomolecule under force F (see the Supporting Material for details on all derivations). The full nonlinear form of Gpull(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, 1/2κS
x‡2(F) << kBT 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 κS/κU(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 (x); the proposed algorithm (Eqs. 3 and 4) removes the contribution of the tether potential Gteth(x) to recover the parameters of the intrinsic ligand-receptor (LR) potential G0(x).The last piece required for Eq. 1, the loading rate , is computed as the time derivative of the force F(t) on the unfolded molecule at its most probable extension at time t: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:(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 κS ≲ κU (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.Multiple relaxation speeds, 2.Folding/binding events at low forces, and 3.A large number of events at each speed.
Open in a separate windowaFixed at value that minimized least-squares error. 相似文献
- 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 κS 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.
Table 1
On-rate and the location and height of the activation barrier from the fit of simulated data to the theory in Eq. 2Folding | k0 (s−1) | x‡ (nm) | ΔG‡ (kBT) | ν |
True | 9.5 × 103 | 2.2 | 2.0 | — |
Fit | 8 ± 2 × 103 | 2.2 ± 0.2 | 1.8 ± 0.5 | 0.54a |
Binding (LTR) | (s−1) | (nm) | (kBT) | ν |
True | 28 | 1.56 | 1.7 | — |
Fit | 24 ± 3 | 1.57 ± 0.09 | 1.8 ± 0.4 | 0.53a |
Binding (LR) | k0 (s−1) | x‡ (nm) | ΔG‡ (kBT) | — |
True | 2.8 | 3.0 | 4.0 | — |
Fit | 2.7 ± 0.2 | 2.9 ± 0.1 | 4.1 ± 0.1 | — |
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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.
Open in a separate windowHere, we use simple kinetic models to elucidate the basic properties of pathway activation by deactivation of a negative regulator (hereafter referred to as mechanism II), as compared with the standard activation of a positive regulator (mechanism I). The analysis is presented in the context of protein phosphorylation, but the conclusions may be generalized to other reversible modifications or to allosteric binding interactions. The common first step is phosphorylation of the regulatory molecule by the kinase. The activity of the upstream kinase such as Akt may be represented by a dimensionless, time (t)-dependent input signal function, s(t). We assume that the total amount of regulator is constant and define its phosphorylated fraction as ϕ(t). Neglecting concentration gradients and saturation of the upstream kinase and of the opposing (constitutively active) phosphatase(s), the conservation of phosphorylated regulator is expressed as follows (see Text S1 in the Supporting Material):(1)The parameter kp is the pseudo-first-order rate constant of protein dephosphorylation. In the case of s = constant (i.e., subject to a step change at t = 0), the properties of this simplified kinetic equation are well known (5) and may be summarized as follows. As the magnitude of the signal strength s increases, the steady-state value of ϕ, ϕss, increases in a saturable fashion; when s >> 1, ϕss approaches its maximum value of 1 and is insensitive to further increases in s. The kinetics of ϕ(t) approaching ϕss become progressively faster as s increases, however.Next, we model the influence of the regulator on a downstream response. Defining the fractional response as ρ and following analogous assumptions as above, we formulate equations for mechanisms I and II as follows:(2)In each equation, the first term on the right-hand side describes activation, and the second, deactivation. In mechanism I, the effective rate constant of activation increases linearly with ϕ, from a minimum value of ka,0 when ϕ = 0 up to a maximum value of ka,max when ϕ = 1; the deactivation rate constant is fixed at kd,0. Conversely, in mechanism II, the effective rate constant of deactivation decreases linearly with ϕ, from a maximum value of kd,0 when ϕ = 0 down to a minimum value of kd,min when ϕ = 1; in this mechanism, the activation rate constant is fixed at ka,0. The initial condition is assigned so that ρ is stationary when ϕ = 0. To further set the two mechanisms on a common basis, we define dimensionless parameters such that the maximum steady-state value of ρ (with ϕss = 1) is the same for both mechanisms I and II,g = ka,max/ka,0 = kd,0/kd,min; K = ka,0/kd,0.(3)With these definitions, each conservation equation is reduced to the following dimensionless form:(4)Mechanisms I and II (Fig. 1
a) are compared first at the level of their steady-state solutions, ρss, for stationary s. Equation 1 yields the familiar hyperbolic dependence of ϕss on s, and ρss(s) has the same shape for both mechanisms. However, whereas ρss of mechanism I shows saturation at a lower value of s than ϕss, the opposite is true of mechanism II (Fig. 1
b). Thus, mechanism II retains sensitivity to the input even while phosphorylation of the upstream regulator shows saturation. This is perhaps more readily seen when ϕss(s) is replaced with a sigmoidal Hill function (i.e., with s replaced by sn in Eq. 1) (Fig. 1
c). The key parameter that affects the relative sensitivities of mechanisms I and II and the disparity between them is the gain constant, g (see Text S1 in the Supporting Material). As this parameter is increased, ρss of mechanism I becomes increasingly saturable with respect to ϕss (Fig. 1
d), whereas ρss of mechanism II gains sensitivity as ϕss approaches 1 (Fig. 1
e). As an illustrative example, consider that when ϕss is increased from 0.90 to 0.95, or from 0.98 to 0.99, the amount of the negative regulator in the active state is reduced by a factor of 2 (see Fig. S2).Open in a separate windowFigure 1Steady-state properties of mechanisms I and II. (a) Schematics of direct (I) and indirect (II) activation. (b) Steady-state dose responses, ρss(s), of mechanisms I and II along with phosphorylation of the upstream regulator, ϕss(s) (Eq. 1 at steady state); K = 0.05, g = 100. (c) Same as panel b, except with a sigmoidal ϕss(s) (Hill function with n = 4). (d) Steady-state output, ρss, of mechanism I vs. ϕss for K = 0.05 and indicated values of the gain constant, g. (e) Same as panel d, but for mechanism II. To see this figure in color, go online.The two mechanisms also show distinct temporal responses. In the response of mechanism I to a step increase in s, ρ(t) approaches ρss with a timescale that generally becomes faster as s increases. Unless the kinetics of ϕ(t) are rate-limiting, the timescale is ∼kd,0–1(1–ρss) (Fig. 2
a; see also Text S1 and Fig. S3 in the Supporting Material). Conversely, the response of mechanism II generally becomes slower as s increases, inasmuch as the frequency of deactivation decreases whereas that of activation is constant, with a timescale of ∼ka,0–1ρss (Fig. 2
b). To approximate a transient input, we model s(t) as a step increase followed by a decay. For mechanism I, the response ρ(t) is such that the variation in the time of the peak, as a function of the step size, is modest. The subsequent decay is prolonged when ϕ(t) hovers close to saturation (Fig. 2
c). Such kinetic schemes have been analyzed in some detail previously (6,7). In contrast, the response of mechanism II to the transient input is such that the system retains sensitivity and consistent decay kinetics beyond the saturation of ϕ(t). The distinctive feature is that ρ(t) peaks noticeably later in time as the magnitude of the peak increases (Fig. 2
d).Open in a separate windowFigure 2Kinetic properties of mechanisms I and II. (a) Response of mechanism I to a step change in s from zero to the indicated s(0). Time is given in units of kpt; parameters are K = 0.05, g = 10, and kd,0 = 0.1kp. (b) Same as panel a, but for mechanism II. (c) Same as panel a, but for a transient input, s(t) = s(0)exp(–0.03kpt). d) Same as panel c, but for mechanism II. To see this figure in color, go online.Having established the basic steady state and kinetic properties of mechanism II as compared with the canonical mechanism I, we considered what outcomes could be achieved by linking these motifs in series or in parallel. Such schemes are identified in the Akt/mTOR signaling network, for example (see Fig. S4). In a standard kinase activation cascade, it is understood that the properties of saturation and sensitivity are compounded with each step of the cascade (8). Thus, two sequential steps of mechanism I yield progressive saturation of the steady-state output at lower s (Fig. 3
a), and the desaturating effect of mechanism II is likewise compounded (Fig. 3
b). By corollary it follows that a sequence of mechanisms I and II will show an intermediate dose response; that is, the mechanism II step offsets the saturation effect of mechanism I.Open in a separate windowFigure 3Serial and parallel schemes incorporating mechanism I or/and II. (a) Steady-state outputs of two response elements, ρ1 and ρ2, activated by mechanism I in series. At each level, K = 0.05, g = 100. (b) Same as panel a, but for mechanism II in series. (c) Incoherent feedforward loop (FFL) in which mechanisms I and II are activated in parallel to activate and inhibit, respectively, the terminal output. For both mechanisms I and II, K = 0.05, g = 100. The parameters for Eq. 5 are α = 2.5, β = 50. To see this figure in color, go online.A more complex scheme is to combine the two mechanisms in parallel, as in an incoherent feedforward loop (FFL) connected to an “AND NOT” output as follows:Output = αρI/(1 + αρI + βρII).(5)Given the differential saturation properties of mechanisms I and II, this scheme readily yields the expected biphasic dose response (9) without the need for disparate values of the parameters (Fig. 3
c). Regarding the kinetics, the analysis shown in Fig. 2 makes it clear that mechanism II naturally introduces time delays in cascades or network motifs. Thus, for the incoherent FFL at high, constant s, activation of inhibition by mechanism II would tend to yield a dynamic response marked by a peak followed by adaptation (see Fig. S5). Analogous calculations were carried out for a coherent FFL as well (see Fig. S6).To summarize our conclusions and their implications for signaling downstream of Akt and other kinases, we have described a distinct, indirect signal transduction mechanism characterized by deactivation of a negative regulator. This motif shows steady-state sensitivity beyond saturation, and therefore the activity of the upstream kinase, such as Akt, can be relatively high. By comparison, the direct activation of signaling by phosphorylation requires that activity of the kinase be regulated, or specifically countered by high phosphatase activity, to maintain sensitivity and avoid saturation of the response. The mechanism described here also introduces relatively slow kinetics (for comparable parameter values). This property, together with its extended range of sensitivity, would allow the motif to be incorporated in signaling networks to yield desired steady and unsteady responses in a robust manner. Considering that key signaling processes mediated by Akt (notably activation of the mammalian target of rapamycin (mTOR) pathway) are achieved by deactivation of negative regulators, we assert that greater recognition of this mechanism and of its distinct properties is warranted. 相似文献
Table 1
Survey of Akt substrates and downstream signalingSubstrate (site) | Effect on substrate | Outcome |
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TSC2 (T1462) | GAP activity ↓ | Rheb, mTOR ↑ |
PRAS40 (T246) | mTOR binding ↓ | mTOR ↑ |
GSK3α/β (S21/S9) | kinase activity ↓ | β-catenin ↑ |
BAD (S136) | Bcl-2/xL binding ↓ | Bcl-2/xL ↑ |
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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 FW (mg axis−1) Electrolyte leakage (%) NO (nmol g−1 FW) LIP(nmol g−1 FW) Control 6.8 ± 0.3 29 ± 2 2.4 ± 0.2 8 ± 1 SNP 10.8 ± 0.6* 20 ± 1* 6.0 ± 0.9* 19 ± 2* Photodegraded SNP 6.6 ± 0.3 27 ± 2 2.5 ± 0.6 9 ± 1 DETA NONOate 9.7 ± 0.9* 18 ± 1* 6.2 ± 0.6* 15.2 ± 0.5*