Biophysical characterization of complexation of DNA with block copolymers of poly(2-dimethylaminoethyl) methacrylate, poly(ethylene oxide), and poly(propylene oxide)

The interactions of DNA (salmon testes) with two new cationic block copolymers made of poly(2-dimethylaminoethyl) methacrylate and poly(ethylene oxide), PEO-pDMAEMA, or poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide), L92-pDMAEMA, were studied with the aim to understand their different in vitro transfection efficiencies when used as nonviral delivery vectors. PEO-pDMAEMA does not show surface activity while L92-pDMAEMA is as surface active as its parent Pluronic L92. Surface tension, titration microcalorimetry, ethidium bromide displacement, and zeta-potential measurements were carried out in phosphate buffers at pH 5 and 7. The association of L92-pDMAEMA with DNA was strongly exothermic at both pHs; the critical aggregation concentration (CAC) corresponded to a N/P ratio of 0.3, the maximum energy evolved was reached for N/P ratios of 0.82 and 1.27 at pH 5 and pH 7, respectively, and the saturation occurred for N/P ratios close to 2. The presence of L92 in the structure of this new block copolymer apparently did not modify the thermodynamic parameters of the interaction with DNA. In contrast, the interaction with PEO-pDMAEMA was significantly less exothermic, and CAC and saturation occurred for N/Ps equal to 0.43 and 1.37, respectively. The strong affinity of L92-pDMAEMA for DNA was reflected in its capacity to displace ethidium bromide and in the jump in the values of the zeta potential when N/P is near 1. Above the N/P ratio at which electroneutral polyplexes are formed, only at pH 5 an excess of L92-pDMAEMA is incorporated in the complexes, resulting in positively charged complexes. The profile of the zeta-potential values obtained for mixtures of L92-pDMAEMA with Pluronic P123 showed a shift to a lower N/P ratio, owing to an easier interaction of L92-pDMAEMA molecules with DNA in the presence of P123. Additionally, a visual inspection of the systems indicates that P123 contributes to stabilize/solubilize the DNA/cationic polymer aggregates, by avoiding the typical phase separation near the charge neutralization point. The information obtained can be particularly useful to optimize the conditions to form efficient polyplexes for gene delivery systems.     

Response probability density functions of strongly non-linear systems by the path integration method

The paper discusses challenges in numerical analysis and numerical/analytical results for strongly non-linear systems—systems with “signum”-type non-linearities. Such non-linearities are implemented for instantaneous variations of the systems’ parameters, to reduce their mean energy response when subjected to random excitations. Numerical results for displacement and velocity response probability density functions (PDFs), energy response PDFs and various order moments are obtained by the path integration technique. Attention is also given to evaluation of mean upcrossing rate, related to the system's half period, via Rice's formula informally applied to discontinuous response PDFs.     

Subharmonic Response of a Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic Excitation

A quasi-isochronous vibroimpact system is considered, i.e. a linear system with a rigid one-sided barrier, which is slightly offset from the system's static equilibrium position. The system is excited by a sinusoidal force with disorder, or random phase modulation. The mean excitation frequency corresponds to a simple or subharmonic resonance, i.e. the value of its ratio to the natural frequency of the system without a barrier is close to some even integer. Influence of white-noise fluctuations of the instantaneous excitation frequency around its mean on the response is studied in this paper. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the application of asymptotic averaging over the period for slowly varying inphase and quadrature responses. The averaged stochastic equations are solved exactly by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset and small random variations of amplitude. Therefore, the analytical results may be expected to be adequate for small values of excitation/system bandwidth ratio or for small intensities of the excitation frequency variations. However, at very large values of the parameter the results are approaching those predicted by a stochastic averaging method. Moreover, Monte-Carlo simulation has shown the moment closure results to be sufficiently accurate in general for any arbitrary bandwidth ratio. The basic conclusion, both of analytical and numerical simulation studies, is a sort of smearing of the amplitude frequency response curves owing to disorder, or random phase modulation: peak amplitudes may be strongly reduced, whereas somewhat increased response may be expected at large detunings, where response amplitudes to perfectly periodic excitation are relatively small.     

On the LQG theory with bounded control

We consider a stochastic optimal control problem in the whole space, where the corresponding HJB equation is degenerate, with a quadratic running cost and coefficients with a linear growth. In this paper we provide full mathematical details on the key estimate relating the asymptotic behavior of the solution as the space variables tend to infinite.     

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the transmitted Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.     

Random Vibrations with Impacts: A Review

Analytical methods and specific results are presented for random vibrations of systems with lumped parameters and “classical” impacts whereby finite relations between impact/rebound velocities are imposed at the impact instants that are not known in advance but rather governed by the equations of motion. Emphasis is placed on the procedures using special piecewise-linear transformation of state variables that exclude the velocity jumps at impacts or makes them small if impact losses are present. In the former case, exact analyses for stationary probability densities of the response to white-noise excitation are possible, whereas the stochastic averaging method is applied in the latter case. Furthermore, the special case of an isochronous system permits a more profound response analysis, such as predicting the spectral density of the response or subharmonic response to narrow-band excitation. The method of direct energy balance is also illustrated, based on direct application of the stochastic differential equation calculus between impacts. Certain two-degree-of-freedom impacting systems are considered, with application to moored systems, as used in ocean engineering.     

Bounded Control of Random Vibration: Hybrid Solution to HJB Equations

Problems of optimal bounded control for randomly excited systems are studied by the Dynamic Programming approach. An effective hybrid solution method has been developed for the corresponding Hamilton–Jacobi–Bellman (HJB) equations for the optimized functional of the response energy. These PDEs are shown to be amenable to an exact analytical solution within a certain outer domain of the phase space. The solution provides boundary conditions for numerical study within the remaining inner domain. The simple dry-friction law had been shown previously to be optimal for a SDOF system within the outer domain and to become asymptotically optimal within the whole phase plane for the important case of so-called long-term control, whereby steady-state response is to be optimized according to the integral cost functional. These results are extended in this paper to MDOF systems by using modal transformation. Thus, the multidimensional dry-friction law is shown to be the optimal one for the case of long-term control. The expected response energy is predicted for this case by a direct energy balance based on application of the SDE Calculus to the energy equation. Stochastic averaging is also used in order to obtain certain reliability estimates. In particular, numerical results for the expected time to first-passage failure are presented, illustrating reduction of reliability due to the imposed bound on the control force. Monte-Carlo simulation results are presented, which demonstrate adequate accuracy of the predictions far beyond the expected applicability range of the asymptotic approaches.