Symmetries in geology and geophysics

Symmetries have played an important role in a variety of problems in geology and geophysics. A large fraction of studies in mineralogy are devoted to the symmetry properties of crystals. In this paper, however, the emphasis will be on scale-invariant (fractal) symmetries. The earth’s topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.

Scaling in geology: landforms and earthquakes.

Landforms and earthquakes appear to be extremely complex; yet, there is order in the complexity. Both satisfy fractal statistics in a variety of ways. A basic question is whether the fractal behavior is due to scale invariance or is the signature of a broadly applicable class of physical processes. Both landscape evolution and regional seismicity appear to be examples of self-organized critical phenomena. A variety of statistical models have been proposed to model landforms, including diffusion-limited aggregation, self-avoiding percolation, and cellular automata. Many authors have studied the behavior of multiple slider-block models, both in terms of the rupture of a fault to generate an earthquake and in terms of the interactions between faults associated with regional seismicity. The slider-block models exhibit a remarkably rich spectrum of behavior; two slider blocks can exhibit low-order chaotic behavior. Large numbers of slider blocks clearly exhibit self-organized critical behavior.

Limited internal friction in the rate-limiting step of a two-state protein folding reaction

Small, single-domain proteins typically fold via a compact transition-state ensemble in a process well fitted by a simple, two-state model. To characterize the rate-limiting conformational changes that underlie two-state folding, we have investigated experimentally the effects of changing solvent viscosity on the refolding of the IgG binding domain of protein L. In conjunction with numerical simulations, our results indicate that the rate-limiting conformational changes of the folding of this domain are strongly coupled to solvent viscosity and lack any significant “internal friction” arising from intrachain collisions. When compared with the previously determined solvent viscosity dependencies of other, more restricted conformational changes, our results suggest that the rate-limiting folding transition involves conformational fluctuations that displace considerable amounts of solvent. Reconciling evidence that the folding transition state ensemble is comprised of highly collapsed species with these and similar, previously reported results should provide a significant constraint for theoretical models of the folding process.

Identification of the proton pathway in bacterial reaction centers: Replacement of Asp-M17 and Asp-L210 with Asn reduces the proton transfer rate in the presence of Cd2+

The reaction center (RC) from Rhodobacter sphaeroides converts light into chemical energy through the reduction and protonation of a bound quinone molecule QB (the secondary quinone electron acceptor). We investigated the proton transfer pathway by measuring the proton-coupled electron transfer, kAB(2) [QA⨪QB⨪ + H+ → QA(QBH)−] in native and mutant RCs in the absence and presence of Cd2+. Previous work has shown that the binding of Cd2+ decreases kAB(2) in native RCs ≈100-fold. The preceding paper shows that bound Cd2+ binds to Asp-H124, His-H126, and His-H128. This region represents the entry point for protons. In this work we investigated the proton transfer pathway connecting the entry point with QB⨪ by searching for mutations that greatly affect kAB(2) (≳10-fold) in the presence of Cd2+, where kAB(2) is limited by the proton transfer rate (kH). Upon mutation of Asp-L210 or Asp-M17 to Asn, kH decreased from ≈60 s−1 to ≈7 s−1, which shows the important role that Asp-L210 and Asp-M17 play in the proton transfer chain. By comparing the rate of proton transfer in the mutants (kH ≈ 7 s−1) with that in native RCs in the absence of Cd2+ (kH ≥ 104 s−1), we conclude that alternate proton transfer pathways, which have been postulated, are at least 103-fold less effective.

Identification of the proton pathway in bacterial reaction centers: Inhibition of proton transfer by binding of Zn2+ or Cd2+

The reaction center (RC) from Rhodobacter sphaeroides converts light into chemical energy through the light induced two-electron, two-proton reduction of a bound quinone molecule QB (the secondary quinone acceptor). A unique pathway for proton transfer to the QB site had so far not been determined. To study the molecular basis for proton transfer, we investigated the effects of exogenous metal ion binding on the kinetics of the proton-assisted electron transfer kAB(2) (QA−•QB−• + H+ → QA(QBH)−, where QA is the primary quinone acceptor). Zn2+ and Cd2+ bound stoichiometrically to the RC (KD ≤ 0.5 μM) and reduced the observed value of kAB(2) 10-fold and 20-fold (pH 8.0), respectively. The bound metal changed the mechanism of the kAB(2) reaction. In native RCs, kAB(2) was previously shown to be rate-limited by electron transfer based on the dependence of kAB(2) on the driving force for electron transfer. Upon addition of Zn2+ or Cd2+, kAB(2) became approximately independent of the electron driving force, implying that the rate of proton transfer was reduced (≥ 102-fold) and has become the rate-limiting step. The lack of an effect of the metal binding on the charge recombination reaction D+•QAQB−• → DQAQB suggests that the binding site is located far (>10 Å) from QB. This hypothesis is confirmed by preliminary x-ray structure analysis. The large change in the rate of proton transfer caused by the stoichiometric binding of the metal ion shows that there is one dominant site of proton entry into the RC from which proton transfer to QB−• occurs.

Stabilization of tetanus and diphtheria toxoids against moisture-induced aggregation.

The progress toward single-dose vaccines has been limited by the poor solid-state stability of vaccine antigens within controlled-release polymers, such as poly(lactide-co-glycolide). For example, herein we report that lyophilized tetanus toxoid aggregates during incubation at 37 degrees C and elevated humidity--i.e., conditions relevant to its release from such systems. The mechanism and extent of this aggregation are dependent on the moisture level in the solid protein, with maximum aggregation observed at intermediate moisture contents. The main aggregation pathway is consistent with formaldehyde-mediated cross-linking, where reactive electrophiles created and stored in the vaccine upon formalinization (exposure to formaldehyde during vaccine preparation) react with nucleophiles of a second vaccine molecule to form intermolecular cross-links. This process is inhibited by the following: (i) succinylating the vaccine to block reactive amino groups; (ii) treating the vaccine with sodium cyanoborohydride, which presumably reduces Schiff bases and some other electrophiles created upon formalinization; and (iii) addition of low-molecular-weight excipients, particularly sorbitol. The moisture-induced aggregation of another formalinized vaccine, diphtheria toxoid, is also retarded by succinylation, suggesting the generality of this mechanism for formalinized vaccines. Hence, mechanistic stability studies of the type described herein may be important for the development of effective single-dose vaccines.