Jerzy Konarski

On the holistic approach in cellular and cancer biology: nonlinearity, complexity, and quasi-determinism of the dynamic cellular network.

A keystone of the molecular reductionist approach to cellular biology is a specific deductive strategy relating genotype to phenotype—two distinct categories. This relationship is based on the assumption that the intermediary cellular network of actively transcribed genes and their regulatory elements is deterministic (i.e., a link between expression of a gene and a phenotypic trait can always be identified, and evolution of the network in time is predetermined). However, experimental data suggest that the relationship between genotype and phenotype is nonbijective (i.e., a gene can contribute to the emergence of more than just one phenotypic trait or a phenotypic trait can be determined by expression of several genes). This implies nonlinearity (i.e., lack of the proportional relationship between input and the outcome), complexity (i.e. emergence of the hierarchical network of multiple cross‐interacting elements that is sensitive to initial conditions, possesses multiple equilibria, organizes spontaneously into different morphological patterns, and is controled in dispersed rather than centralized manner), and quasi‐determinism (i.e., coexistence of deterministic and nondeterministic events) of the network. Nonlinearity within the space of the cellular molecular events underlies the existence of a fractal structure within a number of metabolic processes, and patterns of tissue growth, which is measured experimentally as a fractal dimension. Because of its complexity, the same phenotype can be associated with a number of alternative sequences of cellular events. Moreover, the primary cause initiating phenotypic evolution of cells such as malignant transformation can be favored probabilistically, but not identified unequivocally. Thermodynamic fluctuations of energy rather than gene mutations, the material traits of the fluctuations alter both the molecular and informational structure of the network. Then, the interplay between deterministic chaos, complexity, self‐organization, and natural selection drives formation of malignant phenotype. This concept offers a novel perspective for investigation of tumorigenesis without invalidating current molecular findings. The essay integrates the ideas of the sciences of complexity in a biological context. J. Surg. Oncol. 1998;68:70–78.

ON THE MODIFICATION OF FRACTAL SELF-SPACE DURING CELL DIFFERENTIATION OR TUMOR PROGRESSION

A novel parameter called expansion coefficient has been defined to measure both connectivity and collectivity in a population of cells conquering the available space and self-organizing into tissue patterns of the higher order. Connectivity (i.e. interconnectedness) denotes that there are complex dynamic relationships, not just structural, static ones, in a population of cells enabling the emergence of global features in the system that would never appear in single cells existing out of the system. Collectivity denotes that all interconnected cells interact in a common mode. Evolution of this coefficient during differentiation or tumor progression was investigated by the box-counting method. The population of control or retinoid-treated primary cancer cells cultured in the monolayer (i.e. quasi-2D) system possessed fractal dimension and self-similarity. However, the expansion coefficient was close to zero, indicating that connectivity was low, and no collective state emerged. A significant change of the coefficient occurred when primary cells formed aggregates, quasi-3D systems with increased connectivity, and during treatment of the aggregates with retinoid resulting in a collective state (i.e. in differentiation of cells). Those statistical features were lost during tumor progression. All populations of the secondary cancer cells possessed integer dimension and the expansion coefficient was equal to zero.

SELF-SIMILARITY, COLLECTIVITY, AND EVOLUTION OF FRACTAL DYNAMICS DURING RETINOID-INDUCED DIFFERENTIATION OF CANCER CELL POPULATION

From the reductionist perspective of molecular biology, proliferation or differentiation of eucaryotic cells is a well-defined temporal, spatial, and cell type-specific sequence of molecular cellular events. Some of those events, such as passing of the restriction point in the cell cycle, are of a stochastic nature. Results of this study indicate that, in spite of the intracellular stochasticity, cancer cells can form collective structures with fractal dimension and self-similarity. A transition from the monolayer culture to the aggregated colony facilitated interconnectedness between P19 cells, altered constitutive expression of randomly chosen retinoid-responsive genes, and increased fractal dimension of the entire population. Retinoid-induced emergence of neuron-like phenotype decreased fractal dimension significantly, slowing down dynamics of gene expression. Since the differentiated P19 cells retained both their cancer phenotype and a number of gene defects, we conclude that the appropriate dynamics of intracellular events is neccessary for the proper course of differentiation. Owing to self-similarity, dynamics of cellular expansion can be measured by a fractal dimension in a single cell or in the entire population.

The Gompertzian curve reveals fractal properties of tumor growth

Abstract The normalized Gompertzian curve reflecting growth of experimental malignant tumors in time can be fitted by the power function y(t)=atb with the coefficient of nonlinear regression r⩾0.95, in which the exponent b is a temporal fractal dimension, (i.e., a real number), and time t is a scalar. This curve is a fractal, (i.e., fractal dimension b exists, it changes along the time scale, the Gompertzian function is a contractable mapping of the Banach space R of the real numbers, holds the Banach theorem about the fix point, and its derivative is ⩽1). This denotes that not only space occupied by the interacting cancer cells, but also local, intrasystemic time, in which tumor growth occurs, possesses fractal structure. The value of the mean temporal fractal dimension decreases along the curve approaching eventually integer values; a fact consistent with our hypothesis that the fractal structure is lost during tumor progression.

A luminescene study of Eu(III) and Tb(III) complexes with aminopolycarboxylic acid ligands

Abstract The luminescence of Eu(III) and Tb(III) complexes with eight aminopolycarboxylic acids (APA) in aqueous solution is reported. The ratio of the luminescence intensity of the Eu(III) ion (η=I 615nm /I 593nm ) was analysed and a linear temperature dependence was found. An analysis of the Tb(III) luminescence intensity (λ=545 nm) yielded a linear correlation between the Tb(III) intensity and the hydration number of the Tb—APA complex. On the basis of the luminiscence intensities, a theoritical model has been proposed within the electrostatic dipole approximation. In the model the luminescence intensity of a given electronic transition in dependent on the dipole moment.

On an algebraic approach to the Kratzer oscillator

The ladder operators for the Kratzer‐Fues oscillator have been derived within the algebraic approach. The method is extended to include the rotating Kratzer‐Fues oscillator. For these operators, SU(2) Lie algebra has been constructed. The results obtained differ significantly from those recently derived by Setare and Karimi (2007 Phys. Scr. 75 90‐3). We have shown that in their study the ladder operators and the solutions of the Schrodinger equations with the Kratzer potential have no physical meaning.

On the Gompertzian growth in the fractal space–time

An analytical approach to determination of time-dependent temporal fractal dimension b(t)(t) and scaling factor a(t)(t) for the Gompertzian growth in the fractal space-time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values b(t)(t) and a(t)(t) at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbits tumor growth. The results obtained confirm a possibility of successful mapping of the experimental Gompertz curve onto the fractal power-law scaling function y(t)=a(t)tb(t) and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.

Fractal Structure of Space and Time is Necessary for the Emergence of Self-Organization, Connectivity, and Collectivity in Cellular System

We report that both space and time, in which cell system exists possess fractal structure. This structure emerges owing to non-bijectivity of dynamic cellular network of genes and their regulatory elements. It is lost during tumor progression. This latter state is characterized by damped dynamics of gene expression, loss of connectivity, (i.e., interconnectedness which denotes the existence of complex, dynamic relationships in a population of cells leading to the emergence of global features in the system that would never appear in a single cell existing out of the system), loss of collectivity, (i.e., capability of the interconnected cells to interact in a common mode), and metastatic phenotype. A novel parameter called expansion coefficient has been defined to characterize the observed changes. Fractal structure of both space and time is necessary for cellular system to self-organize.

Assignment of the rovibrational states of deformable diatomic molecules described by different effective potentials

Abstract On the basis of the deformable body model a simple method of assignment of the rovibrational states of diatomic molecules described by the Dunham, Simons—Parr—Finlan and Morse potentials has been considered. The presented approach provides the energy eigenvalues and the corresponding wavefunctions in the exact analytical form. The derived formulae have been used to determine the molecular parameters and to predict the rovibrational spectra of H81Br, 115InD, 12C16O molecules give quite satisfactory reproduction of the experimental data.

Exact solution of the Schrödinger equation with a new expansion of anharmonic potential with the use of the supersymmetric quantum mechanics and factorization method

The study involves finding exact eigenvalues of the radial Schrödinger equation for new expansion of the anharmonic potential energy function. All analytical calculations employ the mathematical formalism of the supersymmetric quantum mechanics. The novelty of this study is underlined by the fact that for the first time the recurrence formulas for rovibrational bound energy levels have been derived employing factorization method and algebraic approach. The ground state and the excited states have been determined by means of the hierarchy of the isospectral Hamiltonians. The Riccati nonlinear differential equation with superpotentials has been solved analytically. It has been shown that exact solutions exist when the potential and superpotential parameters satisfy certain supersymmetric constraints. The results obtained can be utilized both in computations of quantum chemistry and theoretical spectroscopy of diatomic molecules.