Bell's Formula Research Articles

Equilibrium and Non-equilibrium Studies of Protein G using High-Resolution Optical Trapping Unveil Heterogeneous Unfolding Pathways

Recent advances in spatial and temporal resolution of biophysical measurements have revealed ‘beyond two-state’ complexity in protein folding, even for small, single-domain proteins. In this work, the folding/unfolding kinetics of the B1 domain of inmmunoglobulin-binding protein G, a well-studied model system, was investigated using high-resolution optical traps. For non-equilibrium force ramp measurements, simple force-dependent folding distributions were observed. However, multiple peaks were seen in the unfolding force distribution, consistent with multiple parallel unfolding pathways or multiple folded states. We then performed long duration fixed trap measurements while slowly stepping the applied force. We were thereby able to directly measure folding and unfolding rate constants versus force under near equilibrium conditions. All molecules showed the same folding behavior, consistent with a simple two-state Bell's formula. However, these same molecules showed one of two unfolding behaviors at low forces, either ‘fast’ and force independent, or ‘slow’ and strongly force dependent. Equilibrium fixed trap observations of individual molecules for over tens of minutes show no switching between unfolding mechanisms amid hundreds of transitions. However, both unfolding mechanisms are observed for the same individual molecules using the force ramp method, indicating that out of equilibrium pulling may be necessary to switch individual molecule unfolding behavior. Taken together, these measurements indicate that protein G unfolds on a more complex energy landscape containing multiple pathways and intermediate states.

Binding process between E.coli SSB and ssDNA by single-molecule dynamics

Single-stranded DNA binding proteins (SSBs) widely exist in different kinds of creatures. It can bind single-stranded DNA (ssDNA) with high affinity. The binding is sequence independent. SSB can also interact with different kinds of proteins, and thus leading them to work at the special sites. It plays an essential role in cell metabolism. E.coli SSB is a representative of SSB among all kinds of SSBs, it is a homotetramer consisting of four 18.9 kD subunits, the homotetramer is stable under low concentration. E.coli SSB has different binding modes under different salt concentrations (for example NaCl). When NaCl concentration is higher than 200 mM, E.coli SSB can bind 65 nt ssDNA, when NaCl concentration is lower than 20 mM, it can bind 35 nt ssDNA, and when the NaCl concentration is between 20 mM and 200 mM, it can bind 56 nt ssDNA. The characteristics of E.coli SSB are so attractive that a large number of researches have been done to distinguish its binding process. Earlier researchers tried to use stop flow technology to study the interaction between SSB and ssDNA in bulk. However, the high affinity between SSB and ssDNA makes this interaction too rapid to be observed at all, and the dissociate interaction even could not be measured. Single molecule technology which combines with low and accurate force offers researchers another way to achieve this goal. Some researchers observed the unwrapping phenomenon in an optical tweezers pulling experiment. However, they did not find the detailed process of binding or dissociation. In our work, we use a magnetic tweezer to pull the SSB/ssDNA complex and find a special phenomenon like double-state jump. Using the single molecule dynamics to analyse the data, we find that this phenomenon is the combination and dissociation between SSB and ssDNA. After comparing the pulling curve of ssDNA only and SSB/ssDNA complex, we find that the SSB binding process consists of two stages, one is rapid combination/dissociation under the action of a critical force; the other is continuous wrapping following the reduced force. According to Bell formula and SSB/ssDNA complex binding model, we obtain the interaction rate and free energy parameters under 0 pN, and we calibrate the free energy to obtain its continuous wrapping part, so we can obtain the whole free energy landscape and understand the binding process. Our analysis way is also applicable to the case of similar interactions to obtain their interaction details and free energy characteristics.

Heavy‐Atom Tunneling Calculations in Thirteen Organic Reactions: Tunneling Contributions are Substantial, and Bell's Formula Closely Approximates Multidimensional Tunneling at ≥250 K

AbstractMultidimensional tunneling calculations are carried out for 13 reactions, to test the scope of heavy‐atom tunneling in organic chemistry, and to check the accuracy of one‐dimensional tunneling models. The reactions include pericyclic, cycloaromatization, radical cyclization and ring opening, and SN2. When compared at the temperatures that give the same effective rate constant of 3×10−5 s−1, tunneling accounts for 25–95 % of the rate in 8 of the 13 reactions. Values of transmission coefficients predicted by Bell's formula, κBell , agree well with multidimensional tunneling (canonical variational transition state theory with small curvature tunneling), κSCT. Mean unsigned deviations of κBell vs. κSCT are 0.08, 0.04, 0.02 at 250, 300 and 400 K. This suggests that κBell is a useful first choice for predicting transmission coefficients in heavy‐atom tunnelling.

Heavy-Atom Tunneling Calculations in Thirteen Organic Reactions: Tunneling Contributions are Substantial, and Bell's Formula Closely Approximates Multidimensional Tunneling at ≥250 K.

Multidimensional tunneling calculations are carried out for 13 reactions, to test the scope of heavy-atom tunneling in organic chemistry, and to check the accuracy of one-dimensional tunneling models. The reactions include pericyclic, cycloaromatization, radical cyclization and ring opening, and SN 2. When compared at the temperatures that give the same effective rate constant of 3×10-5 s-1 , tunneling accounts for 25-95 % of the rate in 8 of the 13 reactions. Values of transmission coefficients predicted by Bell's formula, κBell , agree well with multidimensional tunneling (canonical variational transition state theory with small curvature tunneling), κSCT . Mean unsigned deviations of κBell vs. κSCT are 0.08, 0.04, 0.02 at 250, 300 and 400 K. This suggests that κBell is a useful first choice for predicting transmission coefficients in heavy-atom tunnelling.

Electro-thermal coupled modeling of PIN diode limiter used in high-power microwave effects simulation

A major failure mode of positive-intrinsic-negative (PIN) diode limiter is the diode junction thermal burnout due to the high-power microwave (HPM) irradiation. Based on the time domain circuit approach, an electro-thermal coupled model including junction transient self-heating effects is presented via analogous between thermal dynamic and electronic circuit topology. Moreover, the proposed model takes into account feedback updating of physical parameters as device junction temperature varying. The diode junction thermal behavior is simulated and verified by the device manufacture measurements. The PIN limiter circuit numerical results of the microwave pulse width dependence on power level under HPM interference validate the well-known semi-empirical Wunsch–Bell relationship. And the turning point value of microwave pulse width obtained in simulations is very close to that specified in Wunsch–Bell formula.

The effect of a mechanical force on quantum reaction rate: Quantum Bell formula

The purpose of this note is to derive a quantum-mechanical analog of Bell's formula, which describes the sensitivity of a chemical reaction to a mechanical pulling force. According to this formula, the reaction rate depends exponentially on the force f, i.e., k(f) ~ exp(f/f(c)), where the force scale f(c) is estimated as the thermal energy k(B)T divided by a distance a between the reactant and transition states along the pulling coordinate. Here I use instanton theory to show that, at low temperatures where quantum tunneling is dominant, this force scale becomes f(c) ~ ℏω/a (in the limit where frictional damping is absent) or f(c) ~ ℏτ(-1)/a (in the strong damping limit). Here ω is a characteristic vibration frequency along the pulling coordinate and τ is a characteristic relaxation time in the reactant state. That is, unlike the classical case where f(c) is unaffected by dissipation, this force scale becomes friction dependent in the quantum limit. I further derive higher-order corrections in the force dependence of the rate, describe generalizations to many degrees of freedom, and discuss connection to other quantum rate theories.

Chemical reactions modulated by mechanical stress: Extended Bell theory

A number of recent studies have shown that mechanical stress can significantly lower or raise the activation barrier of a chemical reaction. Within a common approximation due to Bell [Science 200, 618 (1978)], this barrier is linearly dependent on the applied force. A simple extension of Bell's theory that includes higher order corrections in the force predicts that the force-induced change in the activation energy will be given by -FΔR - ΔχF(2)∕2. Here, ΔR is the change of the distance between the atoms, at which the force F is applied, from the reactant to the transition state, and Δχ is the corresponding change in the mechanical compliance of the molecule. Application of this formula to the electrocyclic ring-opening of cis and trans 1,2-dimethylbenzocyclobutene shows that this extension of Bell's theory essentially recovers the force dependence of the barrier, while the original Bell formula exhibits significant errors. Because the extended Bell theory avoids explicit inclusion of the mechanical stress or strain in electronic structure calculations, it allows a computationally efficient characterization of the effect of mechanical forces on chemical processes. That is, the mechanical susceptibility of any reaction pathway is described in terms of two parameters, ΔR and Δχ, both readily computable at zero force.

Quantum reality: theory and philosophy

Quantum reality: theory and philosophy

Theoretical calculation of Lα X-ray production cross sections of Ga, As, Pt, W and Au atoms by electron impact

The Lα x-ray production cross sections of Ga, As, Pt, W and Au by electron impact are calculated using a modified version of the BELL formula by incorporating both ionic and relativistic corrections in it. The proposed modified Bell model with a single set of parameters can provide an excellent data in the reduced energy range 1≤E/Inl≤106（E and Inl are, respectively, the incident energy and ionization energy） with performance level at least as good as any of the existing methods and models. Experimental data are compared with the present results in cases of with and without the relativistic correction factor GR and the ionic correction factor Fion, with calculated cross sections obtained from the distorted wave Born approximation and with other experimental data available in the literature. The calculated results can be used to simulate the hot-electron energy spectrum and laser plasma yields and accordingly can provide accurate parameters for electron devices.

Theory, analysis, and interpretation of single-molecule force spectroscopy experiments

Dynamic force spectroscopy probes the kinetic and thermodynamic properties of single molecules and molecular assemblies. Here, we propose a simple procedure to extract kinetic information from such experiments. The cornerstone of our method is a transformation of the rupture-force histograms obtained at different force-loading rates into the force-dependent lifetimes measurable in constant-force experiments. To interpret the force-dependent lifetimes, we derive a generalization of Bell's formula that is formally exact within the framework of Kramers theory. This result complements the analytical expression for the lifetime that we derived previously for a class of model potentials. We illustrate our procedure by analyzing the nanopore unzipping of DNA hairpins and the unfolding of a protein attached by flexible linkers to an atomic force microscope. Our procedure to transform rupture-force histograms into the force-dependent lifetimes remains valid even when the molecular extension is a poor reaction coordinate and higher-dimensional free-energy surfaces must be considered. In this case the microscopic interpretation of the lifetimes becomes more challenging because the lifetimes can reveal richer, and even nonmonotonic, dependence on the force.

Electron impact total ionization cross sections of beryllium and boron isoelectronic ions

A simple user-friendly empirical formula is proposed to calculate the electron impact total single ionization cross sections of beryllium-like Be, B+, C2+, N3+, O4+, Ne6+, Fe22+, Hg76+, and U88+ and boron-like B, C+, N2+, O3+, Ne5+, Fe21+, Kr31+, and Xe49+ targets. A simplified and modified version of the Bell formula is used in the present model incorporating the ionic correction factor. The predicted ionization cross sections are compared with the available experimental and theoretical results. Excellent agreements are found for the incident energies ranging from threshold to 106 eV considered herein. The presented results achieve a level of agreement that is better than the calculations from the existing theoretical methods and empirical models. Hence, this model may be an excellent choice, due to its simplicity, for the practitioners in the fields of applied sciences and technologies.

Calculated electron impact K-shell ionization cross sections for atoms

A simple-to-use empirical model for the calculation of electron impact single K-shell ionization total cross sections for 60 atomic targets from H to U (1 ≤ Z ≤ 92) and the energies from threshold to ultra-relativistic range is described. Simplified BELL formula [K.L Bell, H.B. Gilbody, J.G. Hughes, A.E. Kingston, F.J. Smith, J. Phys. Chem. Ref. Data 12 (1983) 891] is used in the proposed model by incorporating in it both the ionic and relativistic corrections. The predicted K-shell ionization cross sections are compared with recently measured experimental data. Better agreement is found for all atoms considered except Xe. Moreover, the calculated results are also compared with other theoretical methods and empirical formulae. The predicted results achieve a level of agreement with both experimental data those are better than the predictions from the existing theoretical methods and empirical models. With the inherent simplicity of its structure, this model may be a prudent selection for the users where fast generation of electron impact single K-shell ionization total cross sections are required.

Extracting Kinetics from Single-Molecule Force Spectroscopy: Nanopore Unzipping of DNA Hairpins

Single-molecule force experiments provide powerful new tools to explore biomolecular interactions. Here, we describe a systematic procedure for extracting kinetic information from force-spectroscopy experiments, and apply it to nanopore unzipping of individual DNA hairpins. Two types of measurements are considered: unzipping at constant voltage, and unzipping at constant voltage-ramp speeds. We perform a global maximum-likelihood analysis of the experimental data at low-to-intermediate ramp speeds. To validate the theoretical models, we compare their predictions with two independent sets of data, collected at high ramp speeds and at constant voltage, by using a quantitative relation between the two types of measurements. Microscopic approaches based on Kramers theory of diffusive barrier crossing allow us to estimate not only intrinsic rates and transition state locations, as in the widely used phenomenological approach based on Bell's formula, but also free energies of activation. The problem of extracting unique and accurate kinetic parameters of a molecular transition is discussed in light of the apparent success of the microscopic theories in reproducing the experimental data.

Electron-impact ionization ofL-shell atomic species

Electron impact ionization cross sections (EIICS) of 30 $L$-shell targets, with open- and closed-shell configurations in the isoelectronic sequences ranging from Li to Ne, are evaluated using the generalized parameters of our recent modification of BELL formula (MBELL) [Haque et al., Phys. Rev. A 73, 012708 (2006)]. Three sets of parameters, one each for the $1s$, $2s$, and $2p$ orbits, provide an excellent account of the experimental EIICS data of atomic targets, neutral and ionic, up to the atomic number $Z=92$ and incident energies up to about $250\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$. In comparison with the quantum mechanical predictions, it is found that the present MBELL cross sections are in better agreement with the experimental results.

Intrinsic Rates and Activation Free Energies from Single-Molecule Pulling Experiments

We present a unified framework for extracting kinetic information from single-molecule pulling experiments at constant force or constant pulling speed. Our procedure provides estimates of not only (i) the intrinsic rate coefficient and (ii) the location of the transition state but also (iii) the free energy of activation. By analyzing simulated data, we show that the resulting rates of force-induced rupture are significantly more reliable than those obtained by the widely used approach based on Bell's formula. We consider the uniqueness of the extracted kinetic information and suggest guidelines to avoid over-interpretation of experiments.

Generalized beable quantum field theory

Generalising the Broglie-Bohm quantum mechanics without wave-function collapse, Bell formulated the “beable quantum field theory”. The local beables n( t) exist independent of observation and, together with the state vector | t〉, specify completely the state of the world at time t. We calculate the most general transition probabilities between different beable configurations, such that the resulting probability P n ( t) for configuration n coincides with that in ordinary quantum field theory. Bell's formula is a special case.

Exactly solvable model for thermal activation and quantum tunneling in Ohmic systems.

Combining Kramers's energy diffusion (at temperature T ) with dissipative quantum tunneling (through a parabolic potential-energy barrier with curvature frequency ${\ensuremath{\omega}}_{b}$ and height ${U}_{b}$), a model for a localized metastable state (with harmonic frequency ${\ensuremath{\omega}}_{0}$) is formulated and solved exactly for its quasiequilibrium distribution \ensuremath{\rho}(E) and its decay rate \ensuremath{\Gamma}. It is shown that \ensuremath{\rho} remains locally extremely close to (although it essentially differs from) the Boltzmann distribution, unless the Ohmic friction coefficient \ensuremath{\lambda} is extremely small (of order \ensuremath{\Gamma}). Excluding this latter possibility, the ensuing decay rate is discussed for various temperature and friction regimes. It is shown to comprise several known results as special cases. First, an extended version of Bell's formula is shown to be valid in the strong-to-moderate friction regime. It involves the recently discovered crossover between thermal hopping and quantum tunneling at temperature ${T}_{0}$=\ensuremath{\Elzxh}\ensuremath{\kappa}${\ensuremath{\omega}}_{b}$/2\ensuremath{\pi}${k}_{B}$ (where \ensuremath{\kappa} is Kramers's viscosity correction factor). At high temperatures and very strong damping the result neatly reduces to Kramers's Smoluchovsky-limit formula. At zero temperature friction strongly suppresses the decay. Second, in the very weak friction regime \ensuremath{\Gamma} is shown to reduce to an extended version of Melnikov's formula. At high temperatures it equals Kramers's small-viscosity result, whereas at zero temperature and zero damping it attains the expected quantum value, the crossover again being at ${T}_{0}$. Third, in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) an extended version of B\"uttiker, Harris, and Landauer's [Phys. Rev. B 28, 1268 (1983)] result is recovered, now valid for any value of the damping. Finally, some remarks are made in relation to recent cryogenic measurements on metastable flux states.