Time and Space Scale

After choosing an adequate model for each different component of the system and integrating them into a final atomistic model that will be simulated, an important issue is the selection of a discretization scheme to implement the computer representation of the ion channel and its environment. Within the framework of a computer experiment, the adjective realistic is strictly related to the phenomena one wants to study, and to the resolution required to reproduce those phenomena. The basic idea for modeling many-body systems is to build a set of rules that apply to each component and let the system evolve dynamically. Ensemble and time averages are then computed to obtain observables that are compared with experiment to validate the model. A characteristic of ion channel systems is that the measurable quantities of direct biological interest evolve in times up to 12 orders of magnitude larger than the smallest atomic or molecular relaxation times (milliseconds versus femtoseconds). In comparison, solid state many-body systems collectively relax in a faster fashion, and the difference between the microscopic 

Analogously, the channel functionality depends on structtual characteristics that extend over a large distance. The interaction of the channel with the membrane is sometimes crucial, both from the structural and electrostatic viewpoint. Furthermore, the structural changes involved in gating are the result of, or are involved directly in, the interaction of the outer protein segments with the membrane. All these facts necessitate the representation of a relatively large system that has to be resolved with angstrom-size accuracy. 

As previously mentioned, a decisive contri bution to the understanding of ion channels has been supplied from experiments. The electrical activity of individual channels is measured both in vivo and in vitro under various conditions.  

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The first step is to define the objectives of the flow model, and to identify those flow aspects that are relevant for the performance of the reactor. Then, the engineer must identify and quantify the various times and space scales involved, as well as the geometry of the system. These actions allow the problem to be represented by a mathematical model. Creating this model accurately is the most crucial task in the flow modeling project. 

Valentini, R., Scarascia Mugnozza, G.E., De Agnelis, P. and Matteucci, G. 1995 Coupling water and carbon metabolism of natural vegetation at integrated time and space scales. Agricultural and Forest Meteorology 73 297-306. 

The appropriate time and space scales are imposed by estimated health effects functions, source and population patterns, data quality and availability, and by the user s information needs. These constraints have led to a wide range of analytical approaches. 

Input Errors. Errors in model input often constitute one of the most significant causes of discrepancies between observed data and model predictions. As shown in Figure 2, the natural system receives the "true" input (usually as a "driving function") whereas the model receives the "observed" input as detected by some measurement method or device. Whenever a measurement is made possible source of error is introduced. System inputs usually vary continuously both in space and time, whereas measurements are usually point values, or averages of multiple point values, and for a particular time or accumulated over a time period. Although continuous measurement devices are in common use, errors are still possible, and essentially all models require transformation of a continuous record into discrete time and space scales acceptable to the model formulation and structure. 

System Representation Errors. System representation errors refer to differences in the processes and the time and space scales represented in the model, versus those that determine the response of the natural system. In essence, these errors are the major ones of concern when one asks "How good is the model ". Whenever comparing model output with observed data in an attempt to evaluate model capabilities, the analyst must have an understanding of the major natural processes, and human impacts, that influence the observed data. Differences between model output and observed data can then be analyzed in light of the limitations of the model algorithm used to represent a particularly critical process, and to insure that all such critical processes are modeled to some appropriate level of detail. For example, a... 

Another classification of model is related to the time and space scales of interest. Ambient air quality standards are stated for measurement averaging periods varying from an hour to a year. However, for computational purposes, it is often necessary to use periods of less than an hour for a typical resolution-cell size in a model. Spatial scales of interest vary from a few tenths of a meter (e.g., for the area immediately adjacent to a roadway) up to hundreds of kilometers (e.g., in simulations that will elucidate urban-rural interactions). Large spatial scales are also warranted when multiday simulations are necessary for even a moderate-sized urban area. Under some climatologic conditions, recirculations can cause interaction of today s pollution with tomorrow s. Typical resolution specifications couple spatial scales with temporal sc es. Therefore, the full matrix of time scales and space scales is not needed, because of the dependence of time scales on space scales. Some typical categories by scale are as follows ... 

The fundamental elements of deterministic models involve a combination of chemical and meteorologic input, preprocessing with data transmission, logic that describes atmospheric processes, and concentration-field output tables or displays. In addition to deterministic models, there are statistical schemes that relate precursors (or emission) to photo-chemical-oxidant concentrations. Models may be classified according to time and space scales, depending on the purposes for which th are designed. 

Figure 1. Time and space scales of algae, patches of algae, zooplankton and fish, A, and various means of remote sensing of biological properties in the oceans, B. Note that ships and buoys give no synopticity. Adapted from Esaias (36) ...

Because of the complexity of hydrated PEMs, a full atomistic modeling of proton transport is impractical. The generic problem is a disparity of time and space scales. While elementary molecular dynamics events occur on a femtosecond time scale, the time interval between consecutive transfer events is usually 3 orders of magnitude greater. The smallest pore may be a few tenth of nanometer while the largest may be a few tens of nanometers. The molecular dynamics events that protons transfer between the water filled pores may have a timescale of 100-1000 ns. This combination of time and spatial scales are far out of the domain for AIMD but in the domain of MD and KMC as shown in Fig. 2. Because of this difficulty, in the models the complexity of the systems is restricted. In fact in many models the dynamics of excess protons in liquid water is considered as an approximation for proton conduction in a hydrated Nation membrane. The conformations and energetics of proton dissociation in acid/water clusters were also evaluated as approximations for those in a Nation membrane.16,19 20 22 24 25... 

Definition of an implementation plan, setting up resources, time, and space scales... 

Sansone F.J., Tribble G.W., Buddemeier R.W. and Andrews C.C. (1988b) Time and space scales of anaerobic diagenesis within a coral reef framework. Proceedings of the Sixth International Coral Reef Congress, Australia 3, 367-372. 

Smith S.D. and Jones E.P.(1986) Isotopoc and micrometeorological ocean CO2 fluxes different time and space scales. J. Geophys. Res. 91,10,529-10,532. 

Table II summarizes the pertinent time and space scales in this problem. Assuming the speed of sound is 105cm/sec, a time-step of about 10-3 sec would be required to resolve the motion of sound waves bouncing across the chamber. Chemical timescales, as mentioned above, are about 10-6 sec. This number may be reduced drastically if the reaction rates or density changes are very fast. It takes a sound wave about 10 3 seconds to cross the 1 meter system and it takes the flame front about one second to cross. We further assume that the flame zone is about 10-2 cm wide and that it takes grid spacings of 10 3 cm to resolve the steep gradients in density and temperature in this flame zone.

We believe that devising a way to handle this difficult problem of strongly coupled multiple time and space scales is the challenge we currently face. 

We believe this is remarkable in explaining a substantial fraction of the geochem-icaUy derived estimate. The difference is likely a function ofinput from Trichodesmium blooms (Carpenter and Capone, 1992 Capone et al., 1996), microbial diazotrophs (Montoya et al., 2004) and symbiotic associations (Carpenter et al., 1999). As noted, all three can contribute intense amounts of nitrogen, although the broader inputs over larger time and space scales are not currently well constrained. 

The way forward will be a fascinating and challenging one. As we have summarized, this is because the biogeochemistry of mercury operates at a variety of time and space scales and in many environmental media. Due to the complexity of the processes and the minute quantities of material often encountered in the environment, future research will also require new hypotheses and new instrumentation. Similarly, and as with so many environmental research elforts, new collaborations among scientific disciplines will be required. 

Although many methods have been developed so far to choose the time delay x and to determine the minimum embedding dimension, there exists no general method, especially for many-body systems where modes associated with hierarchical time and space scales are not necessarily decoupled from one another. The question of which geometrical information at a certain hierarchy on the state space can be reconstructed is not trivial at all for real finite time-series data with finite resolution. 

K.L. Denman, A.E. Gargett (1983). Time and space scales of vertical mixing and advection of phytoplankton in the upper ocean. Limnol. Oceanogr., 28, 801-815. 

Large-scale molecular dynamics simulations are producing information on shock-induced reactions on picosecond (ps) to nanosecond (ns) time scales and approaching micron spatial scales. We describe experiments using ultrafast laser methods to produce experimental data on similar time and space scales to help benchmark the simulations as well as motivate their expansion to larger scales and more complicated materials. 

A molecular description of detonation, particularly initiation, has been pursued for decades, with little success. One difficulty has been obtaining high-quality data at the appropriate length and time scales and with molecular specificity. What are the appropriate time and space scales Detonation waves have tjqjical velocities of 6-8 km/s, or equivalently 6-8 nm/ps. Recent molecular dynamics studies suggest that reactions in shocked energetic materials can occur in times as short as a few ps [1-6]. Energy transfer studies on molecular systems also reveal similar fast time scales [7-11]. Therefore, appropriate spectroscopic probes should have ps or better time resolution. Also, shock rise time measurements with sub-ps resolution require samples with surface uniformity better than 6-8 nm over the probed area. 

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