Approximate analytic solution of a simple system of kinetic equations describing the enzymatic synthesis of nucleic acids

Macroscopic kinetic models describing the process of polymerase chain reaction (PCR) are currently solved only by numerical methods, which hampers the development of effective software algorithms for processing the results of the reaction. This paper considers the application of the homotopy perturbation method for obtaining approximate analytical solution of the simplest system of enzymatic kinetic equations describing the synthesis of nucleic acid molecules during PCR. The resulting approximate analytic solution with high accuracy reproduces the results of a numerical solution of the system in a wide range of ratios of enzyme and substrate concentrations both for the case of a large excess of the substrate over the enzyme and vice versa.     

Spring-mass running: simple approximate solution and application to gait stability

The planar spring-mass model is frequently used to describe bouncing gaits (running, hopping, trotting, galloping) in animal and human locomotion and robotics. Although this model represents a rather simple mechanical system, an analytical solution predicting the center of mass trajectory during stance remains open. We derive an approximate solution in elementary functions assuming a small angular sweep and a small spring compression during stance. The predictive power and quality of this solution is investigated for model parameters relevant to human locomotion. The analysis shows that (i), for spring compressions of up to 20% (angle of attack > or = 60 degree, angular sweep < or = 60 degree) the approximate solution describes the stance dynamics of the center of mass within a 1% tolerance of spring compression and 0.6 degree tolerance of angular motion compared to numerical calculations, and (ii), despite its relative simplicity, the approximate solution accurately predicts stable locomotion well extending into the physiologically reasonable parameter domain. (iii) Furthermore, in a particular case, an explicit parametric dependency required for gait stability can be revealed extending an earlier, empirically found relationship. It is suggested that this approximation of the planar spring-mass dynamics may serve as an analytical tool for application in robotics and further research on legged locomotion.     

Stance phase control of above-knee prostheses: knee control versus SACH foot design

The mobility of above-knee amputees (A/K) is limited, in part, due to the performance of A/K prostheses during the stance phase. Currently stance phase control of most conventional A/K prostheses can only be achieved through leg alignment and choice of the SACH (Solid Ankle Cushioned Heel) foot. This paper examines the role of the knee controller in relation to a SACH foot during the stance phase of level walking. The three-dimensional gait mechanics were measured under two stance phase conditions. In the first set of trials, the amputee used a prosthesis with a conventional knee controller that allowed the amputee to maintain the knee joint in full extension during the stance phase. In the second set of trials, the prosthetic knee, during stance, echoed the modified kinematics of the amputee's sound (intact) knee that had been recorded during the previous sound stance phase. Analysis and interpretation of the data indicate the following: (1) SACH foot design can strongly influence the walking mechanics independent of the knee controller; (2) knee controller design and SACH foot design are mutually interdependent; and (3) normal kinematics imposed on the prosthetic knee does not necessarily produce normal hip kinematics (e.g. reduce the abnormal rise in the prosthetic side hip trajectory). Future research is necessary to explore and exploit the interdependency of prosthetic knee control and foot design.     

广义Lotke-Volterra生态模型的非线性奇摄动近似解

非线性奇摄动问题在国际学术界中是一个重要的研究对象。它涉及到许多学科。在一些生态现象中,原始的研究方法只是采取某些简单观察和统计数据来得到结论。但是它对生态现象的实质的研究达不到效果。近来在国际上提出了研究生态学的动力学方法,即人们首先把它归化为代表它的现象本质的微分方程的模型,然后用数学方法来求解对应的方程,最后研究关于生物和数学理论的动力学方面的规律。目前,非线性摄动问题已经被广泛地研究。许多学者已经研究了一些近似理论。近似求解方法已被发展,包括平均法,边界层法,匹配渐近展开和多尺度法等等。研究非线性广义Lotke-Volterra捕食-被捕食生态模型,一个简单而有效的摄动方法被应用到捕食-被捕食生态模型。提出了捕食-被捕食的一个模型,它是一个微分方程系统,并用小的正参数按幂级数展开未知函数,然后得到关于幂级数的系数的方程,并求出它们的解。于是利用摄动方法得到了原问题解的渐近展开式。得到了它是原模型解是一个好的近似的结论,它是一个解析展开式并且能保持其解析运算。最后,给出了一个对应的例子,它说明得到的解具有很好的精度。     

The frequency response of smooth muscle stiffness during Ca2+-activated contraction

To investigate the mechanism of smooth muscle contraction, the frequency response of the muscle stiffness of single beta-escin permeabilized smooth muscle cells in the relaxed state was studied. Also, the response was continuously monitored for 3 min from the beginning of the exchange of relaxing solution to activating solution, and then at 5-min intervals for up to 20 min. The frequency response (30 Hz bandwidth, 0.33 Hz (or 0.2 Hz) resolution) was calculated from the Fourier-transformed force and length sampled during a 3-s (or 5-s) constant-amplitude length perturbation of increasing-frequency (1-32 Hz) sine waves. In the relaxed state, a large negative phase angle was observed, which suggests the existence of attached energy generating cross-bridges. As the activation progressed, the muscle stiffness and phase angle steadily increased; these increases gradually extended to higher frequencies, and reached a steady state by 100 s after activation or approximately 40 s after stiffness began to increase. The results suggest that a fixed distribution of cross-bridge states was reached after 40 s of Ca2+ activation and the cross-bridge cycling rate did not change during the period of force maintenance.     

Ankle extensor proprioceptors contribute to the enhancement of the soleus EMG during the stance phase of human walking

A rapid plantar flexion perturbation applied to the ankle during the stance phase of the step cycle during human walking unloads the ankle extensors and produces a marked decline in the soleus EMG. This demonstrates that sensory activity contributes importantly to the enhancement of the ankle extensor muscle activation during human walking. On average, the EMG begins to decline approximately 52 ms after the perturbation. In contrast, a rapid dorsi flex ion perturbation produces a group Ia mediated short-latency stretch reflex burst with an onset latency of approximately 36 ms. The transmission of sensory traffic from the foot and ankle was suppressed in 10 subjects by an anaesthetic nerve block produced with local injections of lidocaine hydrochloride. The anaesthetic block had no effect on the stance phase soleus EMG, the latencies of the EMG responses, or the magnitude of the EMG decline following the plantar flexion perturbation. Therefore, it is more likely that proprioceptive afferents, rather than cutaneous afferents, contribute to the background soleus EMG during the late stance phase of the step cycle. The large difference in onset latencies between the short-latency reflex and unload responses suggests that the largest of the active group Ia afferents might not contribute strongly to the background soleus EMG, although it remains to be determined which of the proprioceptive pathways provide the more important contributions.     

The simplest walking model: stability, complexity, and scaling.

We demonstrate that an irreducibly simple, uncontrolled, two-dimensional, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. This model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point-mass at the hip, and infinitesimal point-masses at the feet. The feet have plastic (no-slip, no-bounce) collisions with the slope surface, except during forward swinging, when geometric interference (foot scuffing) is ignored. After nondimensionalizing the governing equations, the model has only one free parameter, the ramp slope gamma. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. The analytic calculations find initial conditions and stability estimates for period-one gait limit cycles. The model exhibits two period-one gait cycles, one of which is stable when 0 < gamma < 0.015 rad. With increasing gamma, stable cycles of higher periods appear, and the walking-like motions apparently become chaotic through a sequence of period doublings. Scaling laws for the model predict that walking speed is proportional to stance angle, stance angle is proportional to gamma 1/3, and that the gravitational power used is proportional to v4 where v is the velocity along the slope.     

Adapting stiffness and attack angle through trial and error to increase self-stability in locomotion

Biological systems are outperforming machines in legged locomoting under almost any conditions. This is partly due to their capability of learning from failure and adapting their control approach and morphological features. This paper proposes an approach that extends the spring-loaded inverted pendulum (SLIP) model with the capability to adapt its attack angle (control) and stiffness (morphology) based on previous locomotion attempts. A set of different update rules, i.e., how this experience is used to adapt, are systematically investigated. The results suggest that modifying either attack angle, or stiffness, or both is beneficial with respect to achieve stable locomotion. Particularly, if the current system configuration (control and morphology) outperforms the previous one, the results suggest that increasing the angle and decreasing the stiffness of the system leads to more stable solutions. Consequently, the basic SLIP model extended by the proposed learning capabilities is able to reach stable locomotion over a much wider range of parameter combinations simply through trial and error.     

Automated polymerase chain reaction in capillary tubes with hot air.

We describe a simple, compact, inexpensive thermal cycler that can be used for the polymerase chain reaction. Based on heat transfer with air to samples in sealed capillary tubes, the apparatus resembles a recirculating hair dryer. The temperature is regulated via thermocouple input to a programmable set-point process controller that provides proportional output to a solid state relay controlling a heating coil. For efficient cooling after the denaturation step, the controller activates a solenoid that opens a door to vent hot air and allows cool air to enter. Temperature-time profiles and amplification results approximate those obtained using water baths and microfuge tubes.     

Assessment of neuromuscular response characteristics at the knee following a functional perturbation.

Previous research indicates that both the extent and timing of muscular activation at the knee can be influenced by muscle activity state, joint angle, weight-bearing status and trunk position. However, little research to date has evaluated protective neuromuscular response times and activation patterns to an imposed perturbation with the knee joint in a functional, weight-bearing stance. Hence, we designed a lower extremity perturbation device to produce a sudden, forward and either internal or external rotation moment of the trunk and femur relative to the weight-bearing tibia. Surface electromyography (EMG) recorded long latency reflex times of the medial and lateral quadriceps, hamstring and gastrocnemius muscles in 64 intercollegiate lacrosse and soccer players in response to both internal and external rotation perturbation. We found the gastrocnemius fired significantly faster that the hamstring, which in turn fired significantly faster than the quadriceps. There was also a significant difference in activation times of the medial and lateral hamstring not found for the quadriceps or gastrocnemius muscles. Our findings confirmed that reactive neuromuscular responses following this functional perturbation differ markedly from those previously reported using seated, partial weight-bearing and/or uni-planar models under relaxed conditions.     

Modeling human postural sway using an intermittent control and hemodynamic perturbations

Ground reaction force during human quiet stance is modulated synchronously with the cardiac cycle through hemodynamics [1]. This almost periodic hemodynamic force induces a small disturbance torque to the ankle joint, which is considered as a source of endogenous perturbation that induces postural sway. Here we consider postural sway dynamics of an inverted pendulum model with an intermittent control strategy, in comparison with the traditional continuous-time feedback controller. We examine whether each control model can exhibit human-like postural sway, characterized by its power law behavior at the low frequency band 0.1–0.7 Hz, when it is weakly perturbed by periodic and/or random forcing mimicking the hemodynamic perturbation. We show that the continuous control model with typical feedback gain parameters hardly exhibits the human-like sway pattern, in contrast with the intermittent control model. Further analyses suggest that deterministic, including chaotic, slow oscillations that characterize the intermittent control strategy, together with the small hemodynamic perturbation, could be a possible mechanism for generating the postural sway.     

A shoe sole-based apparatus and method for randomly perturbing the stance phase of gait: test-retest reliability in young adults

Walking on an irregular surface is associated with an elevated risk for a fall at any age. Yet, relatively little is known about how a human responds to an unexpected underfoot perturbation during gait. This is partly due to the difficulty of generating an intermittent but repeatable, unexpected, underfoot perturbation whose size and location are precisely known. So we developed a shoe sole-embedded apparatus for randomly perturbing the stance phase of gait. Medial and lateral flaps were concealed in the soles of pairs of sandals, along with their actuators. Either flap could be deployed within 400ms in the parasagittal plane under a swing foot; this altered the resulting sagittal and frontal plane orientations of the foot during the next stance phase, whereafter the flap was retracted following toe-off for the rest of that gait trial. We tested six healthy young subjects by randomly presenting a single medial or lateral perturbation in 12 of 30 gait trials. Traditional step kinematic measures were used to evaluate the test-retest reliability of the response to the stimulus at two different walking speeds in 60 randomized trials conducted 1 week apart. The method was effective in systematically inducing an alteration in gait, reproducible across visits, as evidenced by acceptable intraclass correlation coefficients for step width, time and length. We conclude that the apparatus and method has potential for measuring the ability of humans to reject one or more unexpected underfoot perturbations during gait.     

Modeling epithelial cell homeostasis: Assessing recovery and control mechanisms

Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.     

传播的人群生态动力学模型

研究了HIV传播的动力学模型,描述了流行性传染病区域的人群传播规律,特别是利用摄动理论对艾滋病的传播动力学非线性方程作了定量、定性的讨论。     

Structural and biochemical studies of SLIP1–SLBP identify DBP5 and eIF3g as SLIP1-binding proteins

In metazoans, replication-dependent histone mRNAs end in a stem-loop structure instead of the poly(A) tail characteristic of all other mature mRNAs. This specialized 3′ end is bound by stem-loop binding protein (SLBP), a protein that participates in the nuclear export and translation of histone mRNAs. The translational activity of SLBP is mediated by interaction with SLIP1, a middle domain of initiation factor 4G (MIF4G)-like protein that connects to translation initiation. We determined the 2.5 Å resolution crystal structure of zebrafish SLIP1 bound to the translation–activation domain of SLBP and identified the determinants of the recognition. We discovered a SLIP1-binding motif (SBM) in two additional proteins: the translation initiation factor eIF3g and the mRNA-export factor DBP5. We confirmed the binding of SLIP1 to DBP5 and eIF3g by pull-down assays and determined the 3.25 Å resolution structure of SLIP1 bound to the DBP5 SBM. The SBM-binding and homodimerization residues of SLIP1 are conserved in the MIF4G domain of CBP80/20-dependent translation initiation factor (CTIF). The results suggest how the SLIP1 homodimer or a SLIP1–CTIF heterodimer can function as platforms to bridge SLBP with SBM-containing proteins involved in different steps of mRNA metabolism.     

Helical axes of skeletal knee joint motion during running

The purpose of this study was to determine the changes in the axis of rotation of the knee that occur during the stance phase of running. Using intracortical pins, the three-dimensional skeletal kinematics of three subjects were measured during the stance phase of five running trials. The stance phase was divided into equal motion increments for which the position and orientation of the finite helical axes (FHA) were calculated relative to a tibial reference frame. Results were consistent within and between subjects. At the beginning of stance, the FHA was located at the midepicondylar point and during the flexion phase moved 20mm posteriorly and 10mm distally. At the time of peak flexion, the FHA shifted rapidly by about 10-20mm in proximal and posterior direction. The angle between the FHA and the tibial transverse plane increased gradually during flexion, to about 15 degrees of medial inclination, and then returned to zero at the start of the extension phase. These changes in position and orientation of FHA in the knee should be considered in analyses of muscle function during human movement, which require moment arms to be defined relative to a functional rotation axis. The finding that substantial changes in axis of rotation occurred independent of flexion angle suggests that musculoskeletal models must have more than one kinematic degree-of-freedom at the knee. The same applies to the design of knee prostheses, if the goal is to restore normal muscle function.     

Dynamic parameters of the dipole layer formed during pulsed photoemission discharging of a plane vacuum diode

The parameters of the current and dipole moment that arise during the discharging of a plane vacuum diode by a short photoemission pulse are studied analytically and numerically. Electron emission is induced by the ionizing radiation incident obliquely on the cathode. An approximate analytic model is developed that, under certain conditions, makes it possible to reduce the problem to that having a known solution for the normally incident ionizing radiation. In the case of a short pulse, scaling relationships for the parameters of the dipole layer are obtained as functions of the diode parameters (the interelectrode gap and applied voltage) and the angle of incidence of the ionizing radiation.     

Sparse gamma rhythms arising through clustering in adapting neuronal networks

Gamma rhythms (30-100 Hz) are an extensively studied synchronous brain state responsible for a number of sensory, memory, and motor processes. Experimental evidence suggests that fast-spiking interneurons are responsible for carrying the high frequency components of the rhythm, while regular-spiking pyramidal neurons fire sparsely. We propose that a combination of spike frequency adaptation and global inhibition may be responsible for this behavior. Excitatory neurons form several clusters that fire every few cycles of the fast oscillation. This is first shown in a detailed biophysical network model and then analyzed thoroughly in an idealized model. We exploit the fact that the timescale of adaptation is much slower than that of the other variables. Singular perturbation theory is used to derive an approximate periodic solution for a single spiking unit. This is then used to predict the relationship between the number of clusters arising spontaneously in the network as it relates to the adaptation time constant. We compare this to a complementary analysis that employs a weak coupling assumption to predict the first Fourier mode to destabilize from the incoherent state of an associated phase model as the external noise is reduced. Both approaches predict the same scaling of cluster number with respect to the adaptation time constant, which is corroborated in numerical simulations of the full system. Thus, we develop several testable predictions regarding the formation and characteristics of gamma rhythms with sparsely firing excitatory neurons.     

Sulfogalactolipid binding protein SLIP 1: a conserved function for a conserved protein

We have studied the species and tissue expression of the 68kD sulfogalactolipid binding protein SLIP 1, originally detected in the male germ cells of the rat (Lingwood: Can. J. Biochem. Cell Biol., 63:1077-1085, 1985). Our results show that SLIP 1 has been highly conserved during evolution and is found in the testes of all vertebrates tested. In studies in the rat, we have found that SLIP 1 is, however, tissue restricted, being found only in the brain (also a major site of sulfogalactolipid biosynthesis) in addition to the testis. SLIP 1 was also detected in mammalian oocytes. The SLIP 1 species detected in brain and oocytes retain the sulfogalactolipid-binding characteristics of rat testicular SLIP 1, indicating that, in addition to immunological features, the glycolipid-binding function of SLIP 1 is conserved in these tissues.