Measurements of relaxation times in proteins are a valu-simple approach is to measure the efficiency of a sampling pattern applied to several different values of T by the able source of information about molecular motions in biological macromolecules such as proteins (1-3). Relaxation reliability achieved in the worst case. Furthermore, it is easier not to consider individual values of T directly, but times are determined by sampling a relaxing magnetization after various delay times, and it is important to choose these simply optimize the sampling pattern for all T values in the range between the largest ( T max ) and smallest ( T min ) delay times carefully in order to obtain accurate values. Here I describe a strategy for determining the optimal pattern of values considered.
When only two sample points are used, the optimal samsampling times for the system under study.
pling patterns for a range of T values are easily determined The optimal sampling patterns for measuring a single re-(7). For narrow ranges (T max Γ T min ), it is best to place laxation time can be determined using Crame Β΄r-Rao lower one point at zero time, and the second point at 1.11 T GM , bounds (4-6), as described in ( 7), and the essential results are summarized below. The time constant, T, of an exponenwhere T GM Γ
T max T min is the geometric mean of the extreme tially decaying function values of T for the range under consideration. For wider ranges, this second point should be placed at a slightly earlier sampling time, 1.11 T opt . When T max Γ
5 T min , this optimum s(t) Γ
Ae 0 t / T
[1] time is T opt Γ
0.9 T GM , and the worst-case reliability (which occurs at T Γ
T max and T Γ
T min ) is 69% of the best-case can be determined by sampling the intensity at two time reliability (which occurs at T Γ
T opt ). points, t 1 and t 2 . The optimal sampling pattern for T is that With larger numbers of sampling points, the optimal samwhich allows T to be determined with the greatest reliability pling patterns are more complicated. For narrow ranges of (defined as the inverse of the fractional error in T ), and the T, the optimal pattern remains equivalent to the optimal optimal locations for the two sampling points are to place sampling pattern for some value, T opt , which is approxione at the beginning (t 1 Γ
0), and the other at t 2 Γ
1.11 mately equal to, but slightly smaller than, T GM . For wider T. With larger numbers of sample points, the results are ranges, it may be more efficient to place the sample points similar: it is always best to place some of these sample points at several different times, and this can be explored numeriat zero time, and the rest at some optimal time, proportional cally. As the optimization is over two or more variables, it to T. If the number of sample points is very large, the optimal is necessary to use multidimensional search routines. Several pattern is to place 22% of the points at t Γ
0 and 78% at such routines are available (8), but many ''fast'' search t Γ
1.28 T, or more simply and almost as effectively, one routines are unstable when attempting to maximise worstpoint at t Γ
0 and four at t Γ
1.30 T.
case reliabilities, and the best results were achieved with the The choice of optimal sampling times for measuring slow but reliable AMOEBA algorithm (8). Furthermore, it is relaxation times in proteins is more complicated, as differnecessary to perform all numerical calculations using at least ent residues exhibit different relaxation behavior and so double-precision arithmetic, and in some cases, it is neceshave their own optimal sampling patterns. It is not possisary to use quadruple precision. ble to choose an sampling pattern which is optimal for all The results for three sample points are shown in Fig. the different values of T , and it is necessary to choose 1. For narrow ranges, it is best to place one sample point a compromise pattern which is reasonably good for all at zero and the other two points at the same time, 1.19 residues. In this case, the optimal sampling pattern will T opt . For very narrow ranges, T opt Γ T GM , but for wider ranges T opt is slightly smaller as before. When T max Β§ depend on how this compromise is made. A particularly 283