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(6.14) with respect to x gives a Gaussian function, then we realize that with diffusion at an interface the concentration *gradient* looks just like the *concentration* for diffusion from a point source.
The method used to solve this problem can readily be generalized to an arbitrary initial concentration distributio... | {
"Header 1": "Diffusion and Brownian motion",
"token_count": 1990,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Such a system is said to be in a steady state, as defined by the condition that the time derivative of the concentration is zero throughout a region of interest
$$\frac{\partial C}{\partial t} = 0 \tag{6.20}$$
The diffusion equation, Eq. (6.4), now simplifies to the Laplace equation
$$\nabla^2 C = 0 \tag{6.21}$$ ... | {
"Header 1": "Diffusion and Brownian motion",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Now we will focus on this random motion to see what it can tell us about diffusion (Chandrasekhar, 1943).
The molecular approach to diffusion begins with the randomwalk model. A molecule is said to move in discrete steps of fixed length, -, and to take these steps at fixed intervals of time, . We will start with only... | {
"Header 1": "Diffusion and Brownian motion",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Note that $\delta=0.5$ to give x=1 at k=1.
diffusion equation to compare with the random walk because the initial condition $C(x, 0) = \delta(x)$ corresponds to the random walk starting at x = 0.
First, we compare the two results graphically. If we take p = 1/2, then the most likely value of m is N/2. We can no... | {
"Header 1": "Diffusion and Brownian motion",
"token_count": 1836,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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It was noted above that assuming each step is independent leads to the result that the rms distance is proportional to $\sqrt{t}$ . The assumption of independent steps is at the heart of diffusion, and Einstein used it as the basis for a derivation of the diffusion equation. (See Einstein (1956) for a collection of Ei... | {
"Header 1": "Diffusion and Brownian motion",
"Header 2": "6.6 The diffusion equation from microscopic theory",
"token_count": 2029,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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If the frictional coefficient can be measured, or a theory used to derive it, then Eq. (6.62) tells us that we will also know the diffusion constant. So we will now briefly comment on the frictional coefficient of a spherical particle.
The calculation of a frictional coefficient requires solving the equations for vis... | {
"Header 1": "Diffusion and Brownian motion",
"Header 2": "6.6 The diffusion equation from microscopic theory",
"token_count": 2048,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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One reason is that membrane proteins are often attached to cytoplasmic structures and cytoskeletal elements. Another important effect is molecular crowding. Cell membranes often have proteins occupying more than half of their surface area. Membrane proteins tend to get in each other's way, obstructing their random walk... | {
"Header 1": "Diffusion and Brownian motion",
"Header 2": "6.6 The diffusion equation from microscopic theory",
"token_count": 612,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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This chapter will continue the treatment of dynamic processes started in Chapter 6. In that chapter the dynamic process of diffusion was treated as transitions within a continuum of states. Here we will take the opposite extreme. We will look at interconversions between just two states, and model these interconversions... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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The temperature dependence of a rate is often characterized by the $Q_{10}$ , which is the factor by which the rate changes for a 10 degree change in temperature. Starting with Eq. (7.5), the $Q_{10}$ can be related to the activation energy by the following approximate expression
$$E^{\dagger} = \frac{RT^2 \ln(Q... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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(7.5), replacing G by RT ln K, and taking the natural logarithm leads to
$$\ln \alpha = \ln C - E^{\dagger 0}/RT + \Phi \ln K \tag{7.15}$$
Equation (7.15) expresses the linear dependence of the logarithm of the rate constant on the logarithm of the equilibrium constant. This relation has been applied to a wide rang... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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7.10.** Parabolas at $x = \pm 1$ define the potential energy as a function of reaction coordinate for the reactant and product. The intersection defines the transition state with an energy $G^{\dagger}$ . Raising the energy on the product side changes the driving force and alters the position and energy of the trans... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"token_count": 2006,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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(6.64) to express the flux of probability
$$J = D\left(-\frac{P}{kT}\frac{dU}{d\xi} - \frac{dP}{d\xi}\right) \tag{7.27}$$
This equation describes how $P(\xi)$ flows under the influence of the potential energy function as well as the random collisions with solvent molecules and other parts of the reacting molecule... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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We will now derive an equation for Po(t), the open time distribution, by asking what is the relationship between Po(t) and Po(tþ dt). Obviously, if a channel stays open until t þ dt, it must also have stayed open until t. A closure can occur in the time between t and t þ dt, so Po(t þ dt) should be smaller than Po(t). ... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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The manipulations used in Chapter 1 lead to an expression for the extent of unfolding
$$\frac{[P_{\rm u}]}{[P_{\rm f}] + [P_{\rm u}]} = \frac{1}{1 + e^{\Delta G_{\rm u}^{\rm o} - m_{\rm u}[{\rm D}]}} \tag{7.40}$$
This has the same basic form as Eqs. (1.13) and (1.27). Measurements of [P<sub>u</sub>] versus [D] can ... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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For this group of mutants we would have
$$k_{\rm u} = k_{\rm a} {\rm e}^{-\Delta \Delta G_{\rm u}/RT} + k_{\rm b} \tag{7.46} \label{eq:ku}$$
If $k_{\rm b}$ starts off being faster than $k_{\rm a}$ in the wild type protein, then mutants with small $\Delta\Delta G_{\rm u}$ effects will not produce a noticeable ... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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#### **Association kinetics**
Chapter 4 introduced the subject of molecular associations, pointing out their role in the initiation of a host of important biological signaling processes. That chapter focused on thermodynamic aspects of associations, and the factors that influence their strength. This chapter will foc... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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The decay constant (1/) is plotted versus [Ca2<sup>þ</sup> ] (from Kao and Tsien, 1988).
#### 8.3 Diffusion-limited association
The rate of association of two molecules depends on how frequently they collide as they move about randomly in solution. In fact, we can envision two molecules with especially strong affin... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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(7.27)). The collision frequency is the flux (*J* from Eq. (6.64)) times the area of the sphere, $4\pi r^2$ . So we evaluate the total flux through an entire spherical shell as $4\pi r^2 J$
total flux =
$$-4\pi r^2 D_b \left( \frac{C_b(r)}{kT} \frac{\partial U(r)}{\partial r} + \frac{\partial C_b(r)}{\partial r} \r... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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The surrounding flat surface has zero flux, and that imposes the boundary condition that the concentration gradient normal to the surface is zero (Section 6.2.4). In the steady-state solution to the spherical problem the gradients are purely radial, and the concentration is constant within hemispherical shells centered... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Substrate binding is slowed by increasing the solution viscosity, thus satisfying a basic condition of the diffusion limit (Blacklow et al., 1988). The velocities of the subsequent catalytic steps are in the same range as that of the substrate binding step seen with physiological concentrations of substrate. Making the... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Since the encounter with the target is the defining event for outcome (2), the surface of the target is treated as an absorbing boundary.
The random walk version of this problem is difficult, but fortunately there is an equivalent problem in steady-state diffusion that is straightforward. If the molecules are produce... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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(8.34) as
$$\frac{\mathrm{dN}}{\mathrm{dt}} = -4\pi a DC \frac{\mathrm{Ns}}{4a + \mathrm{Ns}} \tag{8.37}$$
This integrates to
$$4a \ln (N/N_0) + N - N_0 = -4\pi aDCt$$
(8.38)
where $N_0$ is the initial number of receptors. Although this transcendental equation cannot be solved for N, it is clear that the time... | {
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"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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Thus, the weak nonspecific binding reduces the number of dimensions from three to one, and this assists the association, much as reducing the dimensionality makes it easier for a ligand to find the receptors on the surface of a cell.
An important requirement of this mechanism is that the protein can move easily along... | {
"Header 1": "Chapter 7",
"Header 2": "**Fundamental rate processes**",
"Header 3": "Chapter 8",
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#### **Multi-state kinetics**
The two-state model of Chapter 7 can be used as a basic building block for more complicated processes. When a system has more than two states, conversions between different pairs can occur, and then the kinetics reflects the aggregate behavior of those various transitions. The time cours... | {
"Header 1": "Chapter 7",
"Header 2": "Chapter 9",
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By extension, adding another state would give us a fourth exponential, and so on. Thus, each time we add
a state to a kinetic model, we add another exponential to the time course. This is a very general and important property of multi-state kinetic systems, which will be examined further below.
#### 9.2 | Initial c... | {
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"Header 2": "Chapter 9",
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It is also worth noting the conditions where the timescales do not separate. When the square root term in Eqs. (9.6a) and (9.6b) is small $((\alpha+\beta+\gamma+\delta)^2 \sim 4\alpha\gamma+4\alpha\delta+4\beta\delta)$ , then $\lambda_1$ and $\lambda_2$ will be similar, and the decay constants will not be easily... | {
"Header 1": "Chapter 7",
"Header 2": "Chapter 9",
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It is extremely unlikely that all concentrations will coincidentally have the values of one of the eigenvectors, and eigenvectors often have negative values making such a state physically impossible. Real solutions are sums of all n fundamental solutions, $\sum_{k=1}^{n} \mathbf{U}_k e^{\lambda_k t}$ . Thus, each spec... | {
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"Header 2": "Chapter 9",
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Kinetic systems often exhibit some form of stability, and can settle into a stable state. Stability means that the concentrations are constant and their time derivatives are zero. This requires that the rate constants obey certain relationships. A system might have a stationary state if the number of moles is conserved... | {
"Header 1": "\\*9.6 | Stationarity, conservation, and detailed balance",
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If the sum of the concentrations of all the different interconverting species is T, then the concentration of species A will be $P_A T$ , where $P_A$ is the probability of a molecule being in state A. As a result the kinetic equations for changes in probability have the same basic form as the equations for changes i... | {
"Header 1": "\\*9.6 | Stationarity, conservation, and detailed balance",
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It is very common in single-channel experiments to observe channel openings in clusters or bursts. A long period with no openings will be followed by many openings in rapid succession, and then another long period with no openings (Fig. 9.3).
A three-state model with two closed states can account for this behavior, p... | {
"Header 1": "9.8 Separation of timescales in single channels: burst analysis",
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A channel gating scheme is represented by a set of differential equations, as exemplified by Eqs. (9.43a) and (9.43b) for the two-open-state model, and these equations can be written in the matrix-vector form of Section 9.4. Our state vector is now $\mathbf{P}_0$ , which represents a set of open probabilities for an a... | {
"Header 1": "9.9 General treatment of single-channel kinetics: state counting",
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The equations for multi-state kinetics are clearly very similar for macroscopic and single-molecule systems. The two realms can be connected explicitly. Consider the kinetic scheme for a channel with multiple open and closed states. We start with the full matrix, $\mathbf{Q}$ , for interconversions between all these s... | {
"Header 1": "\\*9.10 Relation between single-channel and macroscopic kinetics",
"token_count": 1964,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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This leads to expressions for $P_{o1}$ and $P_{o2}$ that can be added together to obtain the open-time distribution
$$P_{o} = \frac{1}{(\alpha + \phi)(\lambda_{1} - \lambda_{2})} \left( -((\alpha + \phi)\lambda_{2} + \alpha\beta + \phi\varepsilon)e^{\lambda_{1}t} + ((\alpha + \phi)\lambda_{1} + \alpha\beta + \phi... | {
"Header 1": "\\*9.10 Relation between single-channel and macroscopic kinetics",
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From A<sub>2</sub>, either subunit can switch, so the rate constant for going from A<sub>2</sub> to AB is 2 $\alpha$ . The rate from AB to B<sub>2</sub> is $\alpha$ , because in AB there is only one subunit that can make the transition. Likewise, the rate constant for going from B<sub>2</sub> to AB is 2 $\beta$ , and ... | {
"Header 1": "\\*9.10 Relation between single-channel and macroscopic kinetics",
"token_count": 2019,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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It can be shown that each pathway from $A_1$ to $A_1$ that crosses the barrier corresponds with one pathway from $A_1$ to $A_{-1}$ ; $A_{-1}$ is the image point of $A_1$ . So the probability of arriving at $A_1$ in the presence of the barrier is the sum of the probabilities of arriving at $A_1$ and $A_{-... | {
"Header 1": "\\*9.10 Relation between single-channel and macroscopic kinetics",
"token_count": 1707,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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How enzymes accelerate biochemical reactions is one of the oldest and most challenging problems in biophysics. An enzyme binds with high specificity to a substrate molecule, chemically modifies it, releases the product, and then repeats the cycle. Without the enzyme, the same chemical reaction can still take place, but... | {
"Header 1": "Enzyme catalysis",
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Without the special environment of an enzyme it takes extreme conditions such as high or low pH to create a species with comparable reactivity.
It is one thing to write out the chemical mechanism used by an enzyme, and quite another to explain how the reaction is actually accelerated. The chemical mechanism is an ess... | {
"Header 1": "Enzyme catalysis",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Setting the above derivative equal to zero, solving for [ES], and using the condition [Et] ¼ [E] ¼ [ES], leads to an expression similar in form to Eq. (10.3)
$$v = \frac{k_{\text{cat}}[E_t][S]}{[S] + (k_{\text{cat}} + k_{-1})/k_1}$$
(10.5)
Note that if kcat<< k 1, then Eq. (10.3) is recovered (with KS¼ k 1/k1).
F... | {
"Header 1": "Enzyme catalysis",
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#### 10.5 | Allosteric enzymes
In the analysis of the steady-state approximation it was noted that more complex mechanisms still lead to the same simple qualitative dependence on substrate concentration. This is because many enzymes have a single substrate binding site. Enzymes with multiple substrate binding sites... | {
"Header 1": "Enzyme catalysis",
"token_count": 2027,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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The energy in the vicinity of the minimum at $\xi = a$ is $U(\xi) = \phi_a(\xi - a)^2$ , and the probability of $\xi$ having a particular value is given by the Boltzmann distribution, $P(\xi) = e^{-U(\xi)/kT}$ . Evaluating the Boltzmann integral for this potential energy function gave $P(a) = \sqrt{\phi_a/\pi kT... | {
"Header 1": "Enzyme catalysis",
"token_count": 2000,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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There are many other examples of comparisons between intramolecular and intermolecular reactions, and for the most part, the rate enhancements approach but rarely exceed $10^5$ – $10^6$ (Bruice, 1970). However, the rate enhancement can be even higher when rotation is restricted, and this can also be taken into accoun... | {
"Header 1": "Enzyme catalysis",
"token_count": 483,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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In spite of the importance of the entropy effects just discussed, one naturally thinks about catalysis in terms of a reduction in the energy of the transition state. This old idea in enzyme catalysis introduced by Pauling in 1946 has guided a great deal of experimental and theoretical work. In general, the rates of rea... | {
"Header 1": "10.10 Reducing $E^{\\dagger}$ : transition state complementarity",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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terminal ribose of the tRNA. These studies revealed two residues which when mutated reduce $k_{\rm cat}$ but do not significantly alter the binding of the substrates tyrosine and ATP (Wells and Fersht, 1986). These residues, threonine 40 and histidine 45, therefore play a role in the e... | {
"Header 1": "Fig. 10.7. Tyrosyl-tRNA synthase catalyzes nucleophilic attack of the tyrosine carboxyl group on the $\\alpha$ phosphate of ATP. Threonine and histidine residues form H-bonds with the $\\gamma$ phosphate, and this distorts the angles of bonds to the $\\alpha$ phosphate to assist in the formation of a p... |
The serine oxyanion, which is a strong nucleophile, should be abundant only at very unphysiological pH.
A weakly reactive H2O molecule will behave more like a strongly reactive OH ion in the presence of a general-base catalyst. A generalbase catalyst is a buffer that pulls a proton away from a water molecule. Ester h... | {
"Header 1": "Fig. 10.7. Tyrosyl-tRNA synthase catalyzes nucleophilic attack of the tyrosine carboxyl group on the $\\alpha$ phosphate of ATP. Threonine and histidine residues form H-bonds with the $\\gamma$ phosphate, and this distorts the angles of bonds to the $\\alpha$ phosphate to assist in the formation of a p... |
*b*-Galactosidase splits disaccharides containing galactose. The enzyme attacks the O-linked carbon atom of galactose using a glutamate side chain carboxyl as a nucleophile. As the bond between the enzyme and galactose forms, the bond with the other sugar is broken. The resulting covalent enzyme–galactose complex is un... | {
"Header 1": "Fig. 10.7. Tyrosyl-tRNA synthase catalyzes nucleophilic attack of the tyrosine carboxyl group on the $\\alpha$ phosphate of ATP. Threonine and histidine residues form H-bonds with the $\\gamma$ phosphate, and this distorts the angles of bonds to the $\\alpha$ phosphate to assist in the formation of a p... |
The serine protease example used at the beginning of this chapter would seem like a perfect illustration of base catalysis (Fig. 10.1). In chymotrypsin, histidine 57 should serve as a base to withdraw the proton from serine 195, thus activating it for nucleophilic attack on the peptide carbonyl. However, the pKs don't ... | {
"Header 1": "10.14 Catalysis in serine proteases and strong H-bonds",
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Carbonic anhydrase catalyzes the reaction shown in Scheme (10E)
$$HCO_3^- \stackrel{\longleftarrow}{\longrightarrow} CO_2 + OH^-$$
(10E)
Unlike most enzymes, both the forward and reverse directions of the reaction catalyzed by carbonic anhydrase are physiologically important. This enzyme assists in the solvation of... | {
"Header 1": "10.15 | Marcus' theory and proton transfer in carbonic anhydrase",
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The biological milieu is a salty aqueous solution. Ions dissolved in water are pushed and pulled by the electrical forces from other ions and charged macromolecules. Charged molecules tend to be surrounded by ions of the opposite sign. These surrounding ions neutralize electrostatic interactions and screen them out. Di... | {
"Header 1": "Ions and counterions",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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The Poisson–Boltzmann equation combines the Poisson equation of electrostatics with the Boltzmann distribution. The Poisson equation expresses the general relation between a charge distribution, $\rho(\mathbf{r})$ , and the electrical potential $\varphi(\mathbf{r})$
$$\nabla^2 \varphi(\mathbf{r}) = -\frac{4\pi}{\va... | {
"Header 1": "11.1 The Poisson-Boltzmann equation and the Debye length",
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That means that $\nabla^2$ should be taken in spherical coordinates (Appendix 6). The Poisson-Boltzmann equation cannot be solved analytically in spherical coordinates, so we use the linearized form (Eq. (11.10))
$$\frac{1}{r}\frac{\mathrm{d}^2(r\varphi(r))}{\mathrm{d}r^2} = \frac{1}{\lambda_{\mathrm{D}}^2}\varphi(... | {
"Header 1": "11.1 The Poisson-Boltzmann equation and the Debye length",
"token_count": 1992,
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(11.11)) gives
$$\rho(r) = \frac{q}{4\pi r \lambda_{\rm D}^2} \left( \frac{e^{-(r-a)/\lambda_{\rm D}}}{1 + a/\lambda_{\rm D}} \right) \tag{11.23}$$
The amount of charge in a shell of thickness dr at r is 4*p*r 2 (r)dr. A graph of 4*p*r 2 (r) is plotted i... | {
"Header 1": "11.1 The Poisson-Boltzmann equation and the Debye length",
"token_count": 1901,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Taking the logarithm (base 10) and using log(e) = 0.434 gives
$$\log[\mathrm{H}^+] + \log\frac{[\mathrm{A}_{i+1}]}{[\mathrm{A}_i]} = \log \mathrm{K}_0 - \frac{0.434 \mathrm{wi}}{\mathrm{RT}(n-i)}(2p+2i+1) \tag{11.28}$$
or
$$pH - log \frac{[A_{i+1}]}{[A_i]} = pK + \frac{0.434wi}{RT(n-i)}(2p+2i+1)$$
(11.29)
To know... | {
"Header 1": "11.1 The Poisson-Boltzmann equation and the Debye length",
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"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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Biological membranes usually have some charge at their surface. This charge arises from the polar head groups of phospholipids, charged amino acids on membrane proteins, and charged sugars on membrane glycoproteins. Membrane surface charge will influence the distribution of dissolved ions, and this is described by the ... | {
"Header 1": "II.4 Gouy-Chapman theory and membrane surface charge",
"token_count": 2020,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
The picture of two opposing layers of charge has lead to the term *ionic double layer* for this kind of charge distribution at a surface.
Now, we return to Eq. (11.38). Integration followed by some rearrangement leads to
$$\frac{x}{\lambda_{\rm D}} = \ln \left( \frac{(e^{e\varphi(x)/2kT} + 1)(e^{e\varphi_0/2kT} - 1... | {
"Header 1": "II.4 Gouy-Chapman theory and membrane surface charge",
"token_count": 1970,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(11.41), but with $\varphi(a)$ in place of $\varphi_0$
$$\sigma_{\rm d} = -\sqrt{\frac{2\varepsilon kT\tilde{A}10^{-3}c}{\pi}}\sinh\left(\frac{-e\varphi(a)}{2kT}\right) \tag{11.55}$$
Equations (11.54) and (11.55) provide relations between the charges of the two layers and the modified surface potential, $\varph... | {
"Header 1": "II.4 Gouy-Chapman theory and membrane surface charge",
"token_count": 2047,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Alternatively, even if surface charge
Fig: 11:7: The voltage seen by a channel in a membrane is the membrane potential, -V, minus the difference in surface potentials at each side. In this sketch a much larger surface potential on the right side than the left side reverses the sign of the membrane potential.
![](_p... | {
"Header 1": "II.4 Gouy-Chapman theory and membrane surface charge",
"token_count": 1965,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
We take the potential at $r_{\zeta}$ , defined as the distance used to compute the zeta potential
$$\zeta = \frac{Q}{\varepsilon r_{\zeta}} \left( \frac{e^{-(r_{\zeta} - a)/\lambda_{D}}}{1 + a/\lambda_{D}} \right)$$
(11.70)
For a large sphere $r_{\zeta} \sim a$ , so the exponential term is $\sim$ 1. Substitution... | {
"Header 1": "II.4 Gouy-Chapman theory and membrane surface charge",
"token_count": 639,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Polyelectrolytes such as DNA and RNA have charge distributed over their entire length. The density of charge is usually quite high so that the electrostatic potential near the surface is substantially larger than kT. The Boltzmann term $e^{-e\varphi/kT}$ can therefore no longer be linearized. The Poisson-Boltzmann eq... | {
"Header 1": "11.9 Polyelectrolyte solutions I. Debye–Hückel screening",
"token_count": 2011,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(11.78) with respect to the concentration of that ion
$$kT \ln \gamma_i = \frac{\partial W}{\partial c_i} = \frac{\partial}{\partial c_i} \left( \frac{q^2}{\varepsilon b} \ln \lambda_D \right)$$
(11.79)
With $\lambda_D$ from Eq. (11.9) we can show that for a binary salt
$$\frac{\partial \lambda_{\rm D}}{\partia... | {
"Header 1": "11.9 Polyelectrolyte solutions I. Debye–Hückel screening",
"token_count": 279,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
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The foregoing analysis of Debye-Hückel screening works only for polyelectrolytes with a low charge density. When a polyelectrolyte has a high charge density it attracts ions so strongly that it gives rise to a unique form of association called *counterion-condensation* (Oosawa, 1971). This can be understood with the ai... | {
"Header 1": "11.10 Polyelectrolyte solutions II. Counterion-condensation",
"token_count": 2048,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
There is a corresponding number of counterions free in solution. If $\xi_{\rm d}$ and $\xi_{\rm s}$ are the charge-density parameters for double- and single-stranded DNA, respectively, then the change in the number of free counterions upon melting is
$$i = \frac{1}{\xi_{\rm s}} - \frac{1}{\xi_{\rm d}} \tag{11.90}... | {
"Header 1": "11.10 Polyelectrolyte solutions II. Counterion-condensation",
"token_count": 2038,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Biological systems often fluctuate more noticeably than typical physical and chemical systems. This reflects the large size of many biological molecules and the small size of cells. The molecular nature of matter gives rise to fluctuations in every imaginable property. These fluctuations may or may not be easy to see, ... | {
"Header 1": "Fluctuations",
"token_count": 1116,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
In the tiny volumes of cells and organelles, the number of molecules can be very small, and fluctuations can make this number deviate substantially from the mean. To quantitate these fluctuations we derive a probability distribution for the number of molecules in a small volume. Consider a very small volume element v w... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"token_count": 2049,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
12.2, we can see that the distributions do indeed get broader as $\overline{m}$ increases. However, the magnitude of the deviations relative to the mean decreases as $\sqrt{\Delta m^2}/\overline{m} = 1/\sqrt{\overline{m}}$ . This property will be used to estimate the threshold number of photons that the human eye ca... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"token_count": 2047,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
no quantum effects), then with three modes of kinetic energy, the total is 3kT/2. Each atom lies in a potential energy well, in which displacements from the minimum are
possible in the x, y, and z directions. This gives another 3kT/2 in potential energy. Thus, a molecule with N atoms will have a mean energy (kinetic ... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"token_count": 2039,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
The energy fluctuations of a protein reflect the independent jittering of its many atoms. Thus, a well-defined structure can still have energy fluctuations of the magnitude implied by Eq. (12.26).
#### 12.6 | Fluctuations in protein ionization
Typical proteins have a large number of ionizable groups. Changing the p... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"token_count": 1970,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
For A we have
$$\overline{N_{a}} = \sum_{N_{a}=0}^{N_{t}} N_{a} \frac{N_{t}!}{N_{a}!(N_{t}-N_{a})!} p_{a}^{N_{a}} (1-p_{a})^{N_{t}-N_{a}} = N_{t} p_{a} = N_{t} \frac{\beta}{\alpha+\beta}$$
(12.40)
where Eq. (12.38a) was used to replace $p_a$ .
For the fluctuations we need to know the mean square as well, so we u... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"token_count": 2030,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
The only other way to get an even number is with an odd number at t and a transition in the interval dt. For the first case we need the probability of no transition when the channel is open. The probability of closing is $\beta dt$ , so the probability of not closing is $1-\beta dt$ . For the second case the channel ... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"token_count": 601,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Dynamic processes can be viewed either in real time or in terms of their characteristic frequencies. The real-time perspective is more direct; the correlation function of the preceding section is easily visualized and an exponential time constant relates directly to the time for a fluctuation to decay. In the frequency... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"Header 2": "12.10 The Wiener–Khintchine theorem",
"token_count": 2045,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
With an inverse Fourier transform of the spectral density one can return to $F_{\rm c}(s)$ . This provides a direct route back and forth between the time and frequency representations of dynamic fluctuations. We will now use the Wiener–Khintchine theorem to look at the power spectrum of a fluctuating two-state system.... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"Header 2": "12.10 The Wiener–Khintchine theorem",
"token_count": 2045,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
$$\frac{\mathrm{d}V}{\mathrm{d}t} = \frac{-I}{C} \tag{12.72}$$
Replacing I with V/R (Ohm's law) gives a first order differential equation in V
$$\frac{\mathrm{d}V}{\mathrm{d}t} = \frac{-V}{RC} \tag{12.73}$$
For an initial value V(0) the solution is
$$V = V(0)e^{-t/RC} (12.74)$$
Equation (12.74) does not hel... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"Header 2": "12.10 The Wiener–Khintchine theorem",
"token_count": 2002,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
To derive the correlation function of this signal we average over the product S(0)S(t)
$$\langle S(\mathbf{0})S(t)\rangle = \left\langle A^2 \int\limits_{-\infty-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} I(x_0, y_0)C(x_0, y_0, \mathbf{0})I(x, y)C(x... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"Header 2": "12.10 The Wiener–Khintchine theorem",
"token_count": 2036,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(6.66)), except with a viscosity for cytoplasm that is 2–3 times higher than that of water. Creatine kinase does not aggregate or interact strongly with other proteins, so it is a good probe of the fluid properties of cytoplasm. By contrast, for tubulin was more than 10 times longer than for creatine kinase. Creatine k... | {
"Header 1": "12.2 Number fluctuations and the Poisson distribution",
"Header 2": "12.10 The Wiener–Khintchine theorem",
"token_count": 283,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
In liquids, molecular fluctuations are the cause of friction. The final section of this chapter will show how the study of fluctuations provides a deeper understanding of friction, and transport processes in general. One normally thinks of friction as something that resists motion and slows things down. But this oversi... | {
"Header 1": "12.14 Friction and the fluctuation—dissipation theorem",
"token_count": 1821,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
A cell expends metabolic energy to transport ions, accumulating some and expelling others so that the concentrations differ between the cell's inside and outside. The cell membrane is more permeable to some of these ions than to others, so ions will move passively down their concentration gradients at different rates. ... | {
"Header 1": "Ion permeation and membrane potential",
"token_count": 2030,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
13.2, with $N^{-w}$ representing a large molecule with a net charge of -w.
The small anions and cations, $B^-$ and $A^+$ , can flow across the membrane but the large anions, $N^{-w}$ are trapped on the compartment a side. Without this polyanion the equilibrium situation would be equal concentrations of the sal... | {
"Header 1": "Ion permeation and membrane potential",
"token_count": 2030,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
If the permeability for one of those ions is important, then the membrane potential will change logarithmically, as predicted by Eq. (13.3). The result of this experiment is shown in Fig. 13.4. We see that varying extracellular $[K^+]$ produces the expected change in membrane potential. Above 30 mM, the change is nea... | {
"Header 1": "Ion permeation and membrane potential",
"token_count": 2049,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(13.13) gives a simple linear relationship between current and voltage
$$I = -\frac{z^2 F^2 cD}{RT} \frac{dV}{dx}$$
(13.14)
We can integrate this equation across the membrane. The value of I is constant so the left side becomes $I\delta$ , where $\delta$ is the membrane thickness. With c constant within the memb... | {
"Header 1": "Ion permeation and membrane potential",
"token_count": 2045,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Different K<sup>+</sup> channels are responsible for the resting potential in other cells. In cerebellar
Fig. 13.6. The data from Fig. 13.4 are well fitted by Eq. (13.22) with $R_{K/Na} = 19$ .
granule neurons a K<sup>þ</sup> channel with two pore-domains is responsible for the high ... | {
"Header 1": "Ion permeation and membrane potential",
"token_count": 1017,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Now we will generalize Eq. (13.21) to obtain an expression for the membrane potential as a function of the concentrations and permeability ratios of all the monovalent ions (Patlak, 1960). We start by writing down the condition of zero net current (as we did for two ions in Section 13.4), but we view each net flux as t... | {
"Header 1": "13.7 | The Goldman–Hodgkin–Katz voltage equation",
"token_count": 2016,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(13.28) and add the flux of pumped ions $J_{\text{pump}}$ (this might be a measured flux or a measured pump current divided by zF). The same analysis that gave Eq. (13.32) now gives
$$\Delta V = -\frac{RT}{F} \ln \left( \frac{\sum_{\text{cations}} P_i c_{ib} + \sum_{\text{anions}} P_i c_{ia} + J_{\text{pump}}}{\sum... | {
"Header 1": "13.7 | The Goldman–Hodgkin–Katz voltage equation",
"token_count": 447,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
A large group of transport proteins couples the movement of two or more molecules across a membrane in a stoichiometric combination. For example, red blood cells have an anion exchange protein that couples the movement of Cl<sup>-</sup> to the movement of HCO<sub>3</sub><sup>-</sup> in the opposite direction. This tran... | {
"Header 1": "13.7 | The Goldman–Hodgkin–Katz voltage equation",
"Header 2": "13.9 Transporters and potentials",
"token_count": 2047,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
We return to Eq. (13.18a), dispense with the subscript for Naþ, and rearrange to give
$$Ie^{zFV/RT} = -zFD\frac{d(ce^{zFV/RT})}{dx}$$
(13.41)
As before, a steady state is assumed with constant current independent of x. We can then integrate with respect to x from one side of the membrane to the other. The integral ... | {
"Header 1": "13.7 | The Goldman–Hodgkin–Katz voltage equation",
"Header 2": "13.9 Transporters and potentials",
"token_count": 1616,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Dividing through by common factors (F2 V/RT), multiplying by e2FV/RT 1, and noting that e2FV/RT 1 ¼ (eFV/RT 1)(eFV/RTþ 1) gives
$$(e^{F\Delta V/RT}+1)\sum_{i}P_{i}(c_{bi}e^{F\Delta V/RT}-c_{ai})+4P_{dc}(c_{bdc}e^{2F\Delta V/RT}-c_{adc})=0 \eqno(13.48)$$
This can be rewritten as a quadratic equation in $e^{F\Delta ... | {
"Header 1": "13.7 | The Goldman–Hodgkin–Katz voltage equation",
"Header 2": "13.9 Transporters and potentials",
"token_count": 2046,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Without this assumption the integral in Eq. (13.43) must be evaluated in another way. Here, we will evaluate the integral by treating the potential energy in the exponential as a voltage drop added to a potential energy barrier. So we add a term U(x) to zFV(x), where U represents the peaked function sketched in Fig. 13... | {
"Header 1": "13.7 | The Goldman–Hodgkin–Katz voltage equation",
"Header 2": "13.9 Transporters and potentials",
"token_count": 1825,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Pure lipid bilayers have extremely low permeabilities to inorganic ions. Adding proteinaceous ion channels can increase the permeability by a factor of more than 10<sup>8</sup> , allowing ions to flow across membranes and produce rapid changes in voltage. One can draw a strong analogy with enzymes. Both ion flow and th... | {
"Header 1": "Ion permeation and channel structure",
"token_count": 2033,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
By contrast, $\gamma$ is a molecular property that tells us something about a channel's structure.
Take a cylindrical channel, with a length equal to the thickness of the membrane, and an area $A = \pi r^2$ (Fig. 14.2). This Ohmic channel is considered the most elementary model for an ion channel (Hille, 1991). A... | {
"Header 1": "Ion permeation and channel structure",
"token_count": 2019,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(14.6) the barrier is exactly in the middle of the membrane, so half the voltage difference is added to the barrier height for one direction of flux and subtracted for the other. Extending the logic of a linear free energy relation to the present situation, we assign the fraction of a voltage drop, $\delta$ , to the p... | {
"Header 1": "Ion permeation and channel structure",
"token_count": 2026,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
A theoretical calculation of this force can be made using electrostatics, treating the water and membrane as continuous media with different dielectric constants. This sort of problem was discussed in Section 2.3. The potential energy is obtained by solving the Laplace equation of electrostatics for an ion positioned w... | {
"Header 1": "Ion permeation and channel structure",
"token_count": 2052,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
The terms now add instead of subtract to give 2U. This large energy accounts for gramicidin's low anion permeability. Finally, divalent cations are also impermeant, and this reflects the dependence on $z^2$ . For a divalent cation, z = 2 and we have $z^2U - zU = 2U$ . The image force thus wins out to produce a barrie... | {
"Header 1": "Ion permeation and channel structure",
"token_count": 363,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Binding sites separated by an energy barrier are a generic feature of the potential energy function of an ion in a channel (Fig. 14.7). These basic features have been inferred again and again in studies of ion permeation. We can thus envision ion movement within a channel as a series of hops over barriers (Fig. 14.8). ... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 1842,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(14.16b) to express $\beta$
$$J = \nu \omega e^{\frac{-(E - (zF\Delta V/2\pi))}{RT}} \frac{(c_a e^{zF\Delta V/RT} - c_b)(e^{zF\Delta V/\pi RT} - 1)}{e^{zF\Delta V/RT} - 1}$$
(14.29)
When $c_a = c_b = c$ the result simplifies to
$$I = z F v \omega c e^{-E/RT} (e^{zF\Delta V/2nRT} - e^{-zF\Delta V/2nRT}) = z F v... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 2040,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
When the $\phi$ s are multiplied together, the Ws in successive terms cancel out. This gives a general expression for the products
$$\prod_{k=j}^{i} \phi_k = e^{-(W_i - W_{j-1})/RT}$$
(14.39)
With the energy on the left side of the channel taken as a reference value of zero ( $W_0 = 0$ ), use of these expression... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 2047,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
The value of p reaches a plateau as c exceeds $1/(v\sum_{i=1}^{n-1}e^{-W_i/RT})$ . In fact, $\sum_{i=1}^{n-1}e^{-W_i/RT}$ is actually a binding constant, so we will call it 1/K. (The resemblance between this sum and a partition function is significant.) Keep in mind as we continue that K depends on the $W_i$ s, the... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 2017,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
If we start with a channel with all n sites occupied by As, and denote this as species $A_n$ , it will be converted to a single-file line of the form $A_{n-1}$ B with a rate s. The reverse process occurs with a rate r. This specifies a differential equation for the rate of change of $[A_n]$
$$\frac{d[A_n]}{dt} = ... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 2043,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
The problem is not in the models but in the paucity of structural information available when the studies were carried out. Recent advances in channel structure have dramatically changed this situation, providing proof for many of the features inferred from barrier models. We will now see how crystal structure has provi... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 2056,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
14.14c. Once this third ion reaches the entrance to the selectivity filter, the ion lodged in the opposite side of the selectivity filter is rapidly expelled to the right by the electrostatic repulsion. The other ion in the selectivity filter assumes the position previously occupied by the expelled ion and the ion in t... | {
"Header 1": "Ion permeation and channel structure",
"Header 2": "14.7 Rate theory for multibarrier channels",
"token_count": 1751,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Cells can have very complex geometries, and when they do the voltage can vary dramatically between different regions. If ionic current flows through a restricted part of a cell's membrane, then the membrane potential at that location will change rapidly, but the membrane potential at distant locations will change more ... | {
"Header 1": "Cable theory",
"token_count": 2047,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
Opening all of those channels suddenly would change the voltage at that spot but the voltage change would spread through a roughly spherical cell within the Maxwell time constant of $\sim\!10^{-9}\,s$ . This is much faster than any relevant biological signaling process. On the other hand, in an axon or dendrite of len... | {
"Header 1": "Cable theory",
"token_count": 1988,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
(15.10) in terms of more fundamental quantities
$$\lambda = \sqrt{\frac{a\rho_{\rm m}}{2\rho_{\rm c}}}\tag{15.13}$$
The fact that $\lambda$ increases with a illustrates another qualitative point of the preceding section that voltage spreads further in a wider cable.
The cable equation brings into focus a few ke... | {
"Header 1": "Cable theory",
"token_count": 2008,
"source_pdf": "datasets/websources/biochem/Cambridge_University_Press_Molecular_and_Cellular_Biophysics.pdf"
} |
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