Formal languages, part theory, and change

A general definition of interpreted formal language is presented. The notion “is a part of" is formally developed and models of the resulting part theory are used as universes of discourse of the formal languages. It is shown that certain Boolean algebras are models of part theory.

With this development, the structure imposed upon the universe of discourse by a formal language is characterized by a group of automorphisms of the model of part theory. If the model of part theory is thought of as a static world, the automorphisms become the changes which take place in the world. Using this formalism, we discuss a notion of abstraction and the concept of definability. A Galois connection between the groups characterizing formal languages and a language-like closure over the groups is determined.

It is shown that a set theory can be developed within models of part theory such that certain strong formal languages can be said to determine their own set theory. This development is such that for a given formal language whose universe of discourse is a model of part theory, a set theory can be imbedded as a submodel of part theory so that the formal language has parts which are sets as its discursive entities.

Synthetic molecular machines for active self-assembly : prototype algorithms, designs, and experimental study

Computer science and electrical engineering have been the great success story of the twentieth century. The neat modularity and mapping of a language onto circuits has led to robots on Mars, desktop computers and smartphones. But these devices are not yet able to do some of the things that life takes for granted: repair a scratch, reproduce, regenerate, or grow exponentially fast–all while remaining functional.

This thesis explores and develops algorithms, molecular implementations, and theoretical proofs in the context of “active self-assembly” of molecular systems. The long-term vision of active self-assembly is the theoretical and physical implementation of materials that are composed of reconfigurable units with the programmability and adaptability of biology’s numerous molecular machines. En route to this goal, we must first find a way to overcome the memory limitations of molecular systems, and to discover the limits of complexity that can be achieved with individual molecules.

One of the main thrusts in molecular programming is to use computer science as a tool for figuring out what can be achieved. While molecular systems that are Turing-complete have been demonstrated [Winfree, 1996], these systems still cannot achieve some of the feats biology has achieved.

One might think that because a system is Turing-complete, capable of computing “anything,” that it can do any arbitrary task. But while it can simulate any digital computational problem, there are many behaviors that are not “computations” in a classical sense, and cannot be directly implemented. Examples include exponential growth and molecular motion relative to a surface.

Passive self-assembly systems cannot implement these behaviors because (a) molecular motion relative to a surface requires a source of fuel that is external to the system, and (b) passive systems are too slow to assemble exponentially-fast-growing structures. We call these behaviors “energetically incomplete” programmable behaviors. This class of behaviors includes any behavior where a passive physical system simply does not have enough physical energy to perform the specified tasks in the requisite amount of time.

As we will demonstrate and prove, a sufficiently expressive implementation of an “active” molecular self-assembly approach can achieve these behaviors. Using an external source of fuel solves part of the the problem, so the system is not “energetically incomplete.” But the programmable system also needs to have sufficient expressive power to achieve the specified behaviors. Perhaps surprisingly, some of these systems do not even require Turing completeness to be sufficiently expressive.

Building on a large variety of work by other scientists in the fields of DNA nanotechnology, chemistry and reconfigurable robotics, this thesis introduces several research contributions in the context of active self-assembly.

We show that simple primitives such as insertion and deletion are able to generate complex and interesting results such as the growth of a linear polymer in logarithmic time and the ability of a linear polymer to treadmill. To this end we developed a formal model for active-self assembly that is directly implementable with DNA molecules. We show that this model is computationally equivalent to a machine capable of producing strings that are stronger than regular languages and, at most, as strong as context-free grammars. This is a great advance in the theory of active self- assembly as prior models were either entirely theoretical or only implementable in the context of macro-scale robotics.

We developed a chain reaction method for the autonomous exponential growth of a linear DNA polymer. Our method is based on the insertion of molecules into the assembly, which generates two new insertion sites for every initial one employed. The building of a line in logarithmic time is a first step toward building a shape in logarithmic time. We demonstrate the first construction of a synthetic linear polymer that grows exponentially fast via insertion. We show that monomer molecules are converted into the polymer in logarithmic time via spectrofluorimetry and gel electrophoresis experiments. We also demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism. This shows the growth of a population of polymers in logarithmic time. We characterize the DNA insertion mechanism that we utilize in Chapter 4. We experimentally demonstrate that we can control the kinetics of this re- action over at least seven orders of magnitude, by programming the sequences of DNA that initiate the reaction.

In addition, we review co-authored work on programming molecular robots using prescriptive landscapes of DNA origami; this was the first microscopic demonstration of programming a molec- ular robot to walk on a 2-dimensional surface. We developed a snapshot method for imaging these random walking molecular robots and a CAPTCHA-like analysis method for difficult-to-interpret imaging data.

Crustal structure in Southern California from array data

Crustal structure in Southern California is investigated using travel times from over 200 stations and thousands of local earthquakes. The data are divided into two sets of first arrivals representing a two-layer crust. The Pg arrivals have paths that refract at depths near 10 km and the Pn arrivals refract along the Moho discontinuity. These data are used to find lateral and azimuthal refractor velocity variations and to determine refractor topography.

In Chapter 2 the Pn raypaths are modeled using linear inverse theory. This enables statistical verification that static delays, lateral slowness variations and anisotropy are all significant parameters. However, because of the inherent size limitations of inverse theory, the full array data set could not be processed and the possible resolution was limited. The tomographic backprojection algorithm developed for Chapters 3 and 4 avoids these size problems. This algorithm allows us to process the data sequentially and to iteratively refine the solution. The variance and resolution for tomography are determined empirically using synthetic structures.

The Pg results spectacularly image the San Andreas Fault, the Garlock Fault and the San Jacinto Fault. The Mojave has slower velocities near 6.0 km/s while the Peninsular Ranges have higher velocities of over 6.5 km/s. The San Jacinto block has velocities only slightly above the Mojave velocities. It may have overthrust Mojave rocks. Surprisingly, the Transverse Ranges are not apparent at Pg depths. The batholiths in these mountains are possibly only surficial.

Pn velocities are fast in the Mojave, slow in Southern California Peninsular Ranges and slow north of the Garlock Fault. Pn anisotropy of 2% with a NWW fast direction exists in Southern California. A region of thin crust (22 km) centers around the Colorado River where the crust bas undergone basin and range type extension. Station delays see the Ventura and Los Angeles Basins but not the Salton Trough, where high velocity rocks underlie the sediments. The Transverse Ranges have a root in their eastern half but not in their western half. The Southern Coast Ranges also have a thickened crust but the Peninsular Ranges have no major root.

Effects of self energy of the ions on the double layer structure and properties at the dielectric interface

Although numerous theoretical efforts have been put forth, a systematic, unified and predictive theoretical framework that is able to capture all the essential physics of the interfacial behaviors of ions, such as the Hofmeister series effect, Jones-Ray effect and the salt effect on the bubble coalescence remain an outstanding challenge. The most common approach to treating electrostatic interactions in the presence of salt ions is the Poisson-Boltzmann (PB) theory. However, there are many systems for which the PB theory fails to offer even a qualitative explanation of the behavior, especially for ions distributed in the vicinity of an interface with dielectric contrast between the two media (like the water-vapor/oil interface). A key factor missing in the PB theory is the self energy of the ion.

In this thesis, we develop a self-consistent theory that treats the electrostatic self energy (including both the short-range Born solvation energy and the long-range image charge interactions), the nonelectrostatic contribution of the self energy, the ion-ion correlation and the screening effect systematically in a single framework. By assuming a finite charge spread of the ion instead of using the point-charge model, the self energy obtained by our theory is free of the divergence problems and gives a continuous self energy across the interface. This continuous feature allows ions on the water side and the vapor/oil side of the interface to be treated in a unified framework. The theory involves a minimum set of parameters of the ion, such as the valency, radius, polarizability of the ions, and the dielectric constants of the medium, that are both intrinsic and readily available. The general theory is first applied to study the thermodynamic property of the bulk electrolyte solution, which shows good agreement with the experiment result for predicting the activity coefficient and osmotic coefficient.

Next, we address the effect of local Born solvation energy on the bulk thermodynamics and interfacial properties of electrolyte solution mixtures. We show that difference in the solvation energy between the cations and anions naturally gives rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The miscibility of the mixture can either increases or decreases depending on the competition between the solvation energy and translation entropy of the ions. The interfacial tension shows a non-monotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations, and decreases approximately as the square root of the salt concentration for dilute solutions, which is in agreement with the Jones-Ray effect observed in experiment.

Next, we investigate the image effects on the double layer structure and interfacial properties near a single charged plate. We show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. The image charge effect is then studied for electrolyte solutions between two plates. For two neutral plates, we show that depletion of the salt ions by the image charge repulsion results in short-range attractive and long-range repulsive forces. If cations and anions are of different valency, the asymmetric depletion leads to the formation of an induced electrical double layer. For two charged plates, the competition between the surface charge and the image charge effect can give rise to like- charge attraction.

Then, we study the inhomogeneous screening effect near the dielectric interface due to the anisotropic and nonuniform ion distribution. We show that the double layer structure and interfacial properties is drastically affected by the inhomogeneous screening if the bulk Debye screening length is comparable or smaller than the Bjerrum length. The width of the depletion layer is characterized by the Bjerrum length, independent of the salt concentration. We predict that the negative adsorption of ions at the interface increases linearly with the salt concentration, which cannot be captured by either the bulk screening approximation or the WKB approximation. For asymmetric salt, the inhomogeneous screening enhances the charge separation in the induced double layer and significantly increases the value of the surface potential.

Finally, to account for the ion specificity, we study the self energy of a single ion across the dielectric interface. The ion is considered to be polarizable: its charge distribution can be self-adjusted to the local dielectric environment to minimize the self energy. Using intrinsic parameters of the ions, such as the valency, radius, and polarizability, we predict the specific ion effect on the interfacial affinity of halogen anions at the water/air interface, and the strong adsorption of hydrophobic ions at the water/oil interface, in agreement with experiments and atomistic simulations.

The theory developed in this work represents the most systematic theoretical technique for weak-coupling electrolytes. We expect the theory to be more useful for studying a wide range of structural and dynamic properties in physicochemical, colloidal, soft-matter and biophysical systems.

Metallogenic Provinces of the Southwestern United States and Northern Mexico

Some of the metallogenic provinces of the southwestern United States and northern Mexico are defined by the geographic distribution of trace elements in the primary sulfide minerals chalcopyrite and sphalerite. The elements investigated include antimony, arsenic, bismuth, cadmium, cobalt, gallium, germanium, indium, manganese, molybdenum, nickel, silver, tellurium, thallium, and tin. Of these elements, cobalt, gallium, germanium, indium, nickel, silver, and tin exhibit the best defined geographic distribution.

The data indicate that chalcopyrite is the preferred host for tin and perhaps molybdenum; sphalerite is the preferred host for cadmium, gallium, germanium, indium, and manganese; galena is the preferred host for antimony, bismuth, silver, tellurium, and thallium; and pyrite is the preferred host for cobalt, nickel, and perhaps arsenic. With respect to the two minerals chalcopyrite and sphalerite, antimony, arsenic, molybdenum, nickel, silver, and tin prefer chalcopyrite; and bismuth, cadmium, cobalt, gallium, germanium, indium, manganese, and thallium prefer sphalerite. This distribution probably is the result of the interaction of several factors, among which are these: the various radii of the elements, the association due to chemical similarities of the major and trace elements, and the degree of ionic versus covalent and metallic character of the metal-sulfur bonds in chalcopyrite and sphalerite. The type of deposit, according to a temperature classification, appears to be of minor importance in determining the trace element content of chalcopyrite and sphalerite.

A preliminary investigation of large single crystals of sphalerite and chalcopyrite indicates that the distribution within a single crystal of some elements such as cadmium in sphalerite and indium and silver in chalcopyrite is relatively uniform, whereas the distribution of some other elements such as cobalt and manganese in sphalerite is somewhat less uniform and the distribution of tin in sphalerite is extremely erratic. The variations in trace element content probably are due largely to variations in the composition of the fluids during the growth of the crystals, but the erratic behavior of tin in sphalerite perhaps is related to the presence of numerous cavities and inclusions in the crystal studied.

Maps of the geographic distribution of trace elements in chalcopyrite and sphalerite exhibit three main belts of greater than average trace element content, which are called the Eastern, Central, and Western belts. These belts are consistent in trend and position with a beltlike distribution of copper, gold, lead, zinc, silver, and tungsten deposits and with most of the major tectonic features. However, there appear to be no definite time relationships, for as many as four metallogenic epochs, from Precambrian to late Tertiary, are represented by ore deposits within the Central belt.

The evidence suggests that the beltlike features have a deep seated origin, perhaps in the sub-crust or outer parts of the mantle, and that the deposits within each belt might be genetically related through a beltlike compositional heterogeneity in the source regions of the ores. Hence, the belts are regarded as metallogenic provinces.