Search

Since 1990, the National Cancer Institute (NCI) has screened more than 60,000 compounds against a panel of 60 human cancer cell lines. The 50-percent growth-inhibitory concentration (GI 50 ) for any single cell line is simply an index of cytotoxicity or cytostasis, but the patterns of 60 such GI 50 values encode unexpectedly rich, detailed information on mechanisms of drug action and drug resistance. Each compound's pattern is like a fingerprint, essentially unique among the many billions of distinguishable possibilities. These activity patterns are being used in conjunction with molecular structural features of the tested agents to explore the NCI's database of more than 460,000 compounds, and they are providing insight into potential target molecules and modulators of activity in the 60 cell lines. For example, the information is being used to search for candidate anticancer drugs that are not dependent on intact p53 suppressor gene function for their activity. It remains to be seen how effective this information-intensive strategy will be at generating new clinically active agents.

Prompted by data from our own experiments on therapy of AIDS, we have taken a fresh look at the problem of analyzing potentiation, synergy, antagonism, enhancement of therapeutic index, and other types of drug interactions (1–3). Initial stages of that inquiry revealed the need for new concepts and new analytical methods that include: 1. Flexible choice of interaction models (clearly, no single type of model can fit all types of interactions). 2. Appropriately robust statistical methods to estimate p-values and confidence intervals (a sound statistical basis is critical in this field, if one is to be confident that there actually is an interaction). 3. Flexibility with respect to experimental design (e.g., checkerboard, constant-ratio, etc.). 4. Statistical techniques for reducing the size of experiments by reducing the number of replicates needed.

Classical enzymology ignores the role which cellular membranes may play in regulating reaction kinetics in vivo. The correct description of cellular metabolic processes must derive from a mass balance equation for each reacting species. An elementary mathematical model, called the “continuous flow stirred tank reactor” in the chemical engineering literature, has been applied to Michaelis-Menten kinetics, substrate inhibition kinetics, and kinetics involving hydrogen ion as a hyproduct. A number of remarkable phenomena, including multiple stationary states, threshold effects, temporal patterns, homeostatic regulation, amplification, and irreversible differentiation can result. Predictions of the model are in qualitative accord with experimental and theoretical studies of insolubilized enzymes, which are conventionally modeled by a more difficult mathematical formalism.

In coenzyme Q-cycles, it is proposed that one electron from the quinol reduces the Rieske iron sulfur center (Em approximately 280 mV) and the remaining electron on the semiquinone reduces cytochrome br (Em approximately -60 mV). The Em for the two-electron oxidation of the quinol is approximately 60 mV and therefore the reduction of cytochrome bT by quinol is not favorable. As the stability constant for the dismutation of the semiquinone decreases, the calculated Em for the Q/QH. couple is lowered to values below the Em of cytochrome bT. Contemporary coenzyme Q-cycles are based on the belief that the lower value for the Em of the Q/QH. couple compared to the Em for cytochrome bT means that the semiquinone is a spontaneous reducing agent for the b-cytochrome. The analysis in the paper shows that this is not necessarily so and that neither binding sites nor ionization of the semiquinone per se alters this situation. For a Q-cycle mechanism to function, ad hoc provisions must be made to drive the otherwise unfavorable reduction of cytochrome bT by the semiquinone or for the simultaneous transfer of both electrons to cytochrome bT and cytochrome c1 (or the iron sulfur protein). Q-cycle mechanisms with these additional provisions can explain the observation thus far accumulated. A linear path which is functionally altered by conformational changes may also explain the data.

The physical principles that material and charge are conserved provide a basis for the design of a membrane capable of performing active ion transport in which the connection between “metabolic energy” input and the ion transport process itself is electrical rather than material. Molecular interactions between components in this system are irrelevant to its function. A second model built on the same principles performs active ion transport in a statically symmetrical membrane. The basis of its operation is a weakly stable unsymmetrical concentration profile arising from an enzyme-catalyzed reaction occurring within the membrane. The function of this membrane is irreversibly terminated (“killed”) by interference either with intramembrane concentration gradients or with the reactions which maintain them. Hence, any attempt to study this system by breaking it apart destroys the basis of its function. The existence of these models reveals the logical insufficiency of the molecular biologist's approach to understanding the basis of active transport: Neither the experimental techniques for structure determination (disruption, purification, characterization, and reconstitution) nor the fundamental question of that discipline (What is the molecular connection between &*|… ?) of the molecular biologist are applicable to the study or interpretation of these model systems. While the model systems are artificial, they incorporate only widely applicable concepts of physical chemistry and biochemical kinetics. The is no reason for excluding such mechanisms in natural membrane transport systems. If they are present, then more effective strategies of investigation will be required.

The major graphical approaches to kinetics—Hill diagrams and the King–Altman rules—can be considerably simplified using network techniques. An algorithmic procedure is developed and applied to several concrete examples which permits systems described in terms of first order transitions to be represented as connected networks. The utility of this representation is: (1) it permits powerful results from network thermodynamics and network theory to be applied to complex kinetic networks such as those representing biological dynamic systems. (2) The intrinsic role played by the thermodynamic principle of microscopic reversibility in networks containing cycles is emphasized: microscopic reversibility implies Onsager reciprocity for perturbed equilibrium systems. It now becomes clear that Onsager reciprocity is a topological property of systems to which thermodynamics is applicable. Reciprocity, or its lack can be rigorously established for macroscopic systems far outside the domain of Onsager’s original demonst...

Kauffman, Shymko & Trabert (1978) have presented a model for sequentialcompartment formation in Drosophila wing disks based upon reactiondiffusion instabilities on an elliptical model for the wing disk. Using numerical methods we have explored the properties of this model using a more accurate representation of the shape of the wing disk and other embryonic structures. The nodal lines of successive eigenfunctions on these domains differ significantly both in form and order from the compartmental lines experimentally observed on the wing disk and are very sensitive to the shape of the domain. The model of Kauffman and coworkers is linear. We present a particular non-linear reaction-diffusion model which produces different patterns. It appears to be characteristic of non-linear models that a priori prediction of the sequence of patterns resulting for given parameters is impossible. The patterns generated by these instability models are generally quite sensitive to changes in the shape of the domain and physical-chemical parameter values. This sensitivity makes it difficult to imagine how pattern form and sequence could be maintained in the face of normal biological variation. We conclude that the use of instability models in this aspect of morphogenesis is an interesting speculation, but one whose validity will be difficult to demonstrate convincingly.

Spectral power densities and autocorrelation functions have been computed for several types of discrete dynamical systems: both chaotic and noisy. In some cases, chaotic dynamical systems can be distinguished from other types of dynamical systems by their appearance in the frequency domain.

Charge-mosaic membranes are currently being considered for a number of practical applications, most notably “piezodialysis“ desalination. In Part I of this series the properties of the charge-mosaic are subjected to a nonequilibrium thermodynamic analysis, with emphasis on the role of the electrical resistance in the solutions bathing the membrane. Four regimes of operation are delineated by the analysis: (i) membrane control, (ii) solution control, (iii) co-ion leakage control, and (iv) polarization control. For a charge-mosaic operating in the regime of membrane control, the “complete” phenomenological coefficients are essentially concentration-independent, hence the flows may be expressed as linear functions of the global forces. If the membrane operates instead in the regime of solution control, the relationships are not linear but the coefficients are still directly calculable.

Attempts to apply traditional techniques of enzyme kinetic analysis(Lineweaver-Burk and Eadie-Hofstee plots) to enzymes compartmentalized by the membranes of cells, organelles, or vesicular membrane fragments will generally lead to incorrect estimates of the kinetic constants of the enzymes and incorrect conclusions about the mechanism of reaction. The error is the consequence of concentration differences, arising through the reaction process, between the solution outside the cell or organelle membrane and the solution in the interior where the reaction is taking place. The use of enzymological plots to interpret data from membrane transport studies, for example, may therefore be misleading. The specific form of the modifications to be expected in these plots for the most common types of reaction mechanisms is presented in the text in graphical form, with the algebraic expressions summarized in an appendix. The plots show curvatures which may be incorrectly interpreted as implying co-operative kinetics when, in fact, they are quite pedestrian. In some simple cases, there exist alternative ways of plotting the data which permit the correct evaluation of kinetic constants and distinction among mechanisms. The effect of cellular compartmentation on inhibition studies is also described.

Using the principle of material balance, a macroscopic concept free of molecular speculations, we can construct very simple models which perform functions equivalent to the biologically important phenomena of facilitated transport, chemiosmosis, and active transport. Rather than being esoteric consequences of special molecular assumptions, each of these phenomena is shown to be necessary consequences of reaction and transport processes coupled by conservation of material. Global properties of the system, such as its symmetry and whether the flow of a species across the system results in net chemical conversion of the reacting species determine which of the phenomena will be observed. Facilitated transport is the expected consequence of the globally conserved flow of a reacting species across a symmetrical membrane, independent of the way in which transport occurs and the position or number of reactions in which it participates during transport. Chemiosmosis is anticipated as a consequence of chemical reaction in an asymmetrical membrane separating reservoirs at different concentrations. Active transport is the expected consequence of simultaneous chemical reaction and transport of a globally conserved reacting species in an asymmetrical membrane when there is net reaction involving some other species. Model systems exemplifying these conclusions are presented here using constitutive equations for reaction and transport which are linear. The conclusions are sufficiently robust, however, that they are independent of structural details by which the symmetry or asymmetry of the membrane is achieved, and appear to remain applicable in the non-linear regime. The formulation developed here, although patterned after non-equilibrium thermodynamics, is not subject to the many inadequacies of that discipline. We state the theorem analogous to the Curie principle in which symmetry occupies role played by isotropy in the thermodynamic version. The existence of coupling between flows and non-conjugate forces follows from the principle of material balance, requiring no ad hoc assumptions. This coupling does not obey Onsager reciprocity except asymptotically near thermodynamic equilibrium.

The collective kinetic behavior of a linear array of cells containing a substrate inhibited enzyme is studied with a model in which each cell is considered a well-stirred compartment surrounded by a semipermeable membrane. At large values of a reaction-permeation modulus, wherein substrate access to interior cells is limited, plots of total reaction rate versus concentration of the external reservoir show sharp projections which correspond to dominant reaction occurring in a pair of symmetrically placed cells. Over prescribed ranges of reservoir concentration, multiple stable steady states can occur, some of which are characterized by asymmetric profiles of concentration and reaction rate across the array. A simple stability criterion is proposed and applied to arrays with arbitrary numbers of cells.