The darwin (d) is a unit of evolutionary change, defined by J. B. S. Haldane in 1949. [1] One darwin is defined to be an e-fold (about 2.718) change in a trait over one million years. Haldane named the unit after Charles Darwin.
The equation for calculating evolutionary change in darwins () is:
where and are the initial and final values of the trait and is the change in time in millions of years. An alternative form of this equation is:
Since the difference between two natural logarithms is a dimensionless ratio, the trait may be measured in any unit. Inexplicably, Haldane defined the millidarwin as 10−9 darwins, despite the fact that the prefix milli- usually denotes a factor of one thousandth (10−3). [2]
The measure is most useful in palaeontology, where macroevolutionary changes in the dimensions of fossils can be compared. Where this is used it is an indirect measure as it relies on phenotypic rather than genotypic data. Several data points are required to overcome natural variation within a population. The darwin only measures the evolution of a particular trait rather than a lineage; different traits may evolve at different rates within a lineage. The evolution of traits can however be used to infer as a proxy the evolution of lineages.
In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 that the van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining rate of chemical reactions and for calculation of energy of activation. Arrhenius provided a physical justification and interpretation for the formula. Currently, it is best seen as an empirical relationship. It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions. The Eyring equation, developed in 1935, also expresses the relationship between rate and energy.
Fitness is the quantitative representation of individual reproductive success. It is also equal to the average contribution to the gene pool of the next generation, made by the same individuals of the specified genotype or phenotype. Fitness can be defined either with respect to a genotype or to a phenotype in a given environment or time. The fitness of a genotype is manifested through its phenotype, which is also affected by the developmental environment. The fitness of a given phenotype can also be different in different selective environments.
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:
In physics, chemistry and biology, a potential gradient is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux.
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
The Tsiolkovsky rocket equation, classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum.
In the theory of evolution and natural selection, the Price equation describes how a trait or allele changes in frequency over time. The equation uses a covariance between a trait and fitness, to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the frequency of alleles within each new generation of a population. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton's work on kin selection. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications in economics.
In biochemistry and pharmacology, the Hill equation refers to two closely related equations that reflect the binding of ligands to macromolecules, as a function of the ligand concentration. A ligand is "a substance that forms a complex with a biomolecule to serve a biological purpose", and a macromolecule is a very large molecule, such as a protein, with a complex structure of components. Protein-ligand binding typically changes the structure of the target protein, thereby changing its function in a cell.
Flory–Huggins solution theory is a lattice model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.
The Eyring equation is an equation used in chemical kinetics to describe changes in the rate of a chemical reaction against temperature. It was developed almost simultaneously in 1935 by Henry Eyring, Meredith Gwynne Evans and Michael Polanyi. The equation follows from the transition state theory, also known as activated-complex theory. If one assumes a constant enthalpy of activation and constant entropy of activation, the Eyring equation is similar to the empirical Arrhenius equation, despite the Arrhenius equation being empirical and the Eyring equation based on statistical mechanical justification.
The Van 't Hoff equation relates the change in the equilibrium constant, Keq, of a chemical reaction to the change in temperature, T, given the standard enthalpy change, ΔrH⊖, for the process. It was proposed by Dutch chemist Jacobus Henricus van 't Hoff in 1884 in his book Études de Dynamique chimique.
Metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and species concentrations, depend on network parameters. In particular it is able to describe how network dependent properties, called control coefficients, depend on local properties called elasticities.
The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.
In laser physics, gain or amplification is a process where the medium transfers part of its energy to the emitted electromagnetic radiation, resulting in an increase in optical power. This is the basic principle of all lasers. Quantitatively, gain is a measure of the ability of a laser medium to increase optical power.
In electrochemistry, the Butler–Volmer equation, also known as Erdey-Grúz–Volmer equation, is one of the most fundamental relationships in electrochemical kinetics. It describes how the electrical current through an electrode depends on the voltage difference between the electrode and the bulk electrolyte for a simple, unimolecular redox reaction, considering that both a cathodic and an anodic reaction occur on the same electrode:
Diffusion is the net movement of anything generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition.
Thermal resistance is a heat property and a measurement of a temperature difference by which an object or material resists a heat flow. Thermal resistance is the reciprocal of thermal conductance.
In astrophysics, the virial mass is the mass of a gravitationally bound astrophysical system, assuming the virial theorem applies. In the context of galaxy formation and dark matter halos, the virial mass is defined as the mass enclosed within the virial radius of a gravitationally bound system, a radius within which the system obeys the virial theorem. The virial radius is determined using a "top-hat" model. A spherical "top hat" density perturbation destined to become a galaxy begins to expand, but the expansion is halted and reversed due to the mass collapsing under gravity until the sphere reaches equilibrium – it is said to be virialized. Within this radius, the sphere obeys the virial theorem which says that the average kinetic energy is equal to minus one half times the average potential energy, , and this radius defines the virial radius.
Laser linewidth is the spectral linewidth of a laser beam.
In statistical mechanics, the mean squared displacement is a measure of the deviation of the position of a particle with respect to a reference position over time. It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system "explored" by the random walker. In the realm of biophysics and environmental engineering, the Mean Squared Displacement is measured over time to determine if a particle is spreading solwly due to diffusion, or if an advective force is also contributing. Another relevant concept, the Variance-Related Diameter, is also used in studying the transportation and mixing phenomena in the realm of environmental engineering. It prominently appears in the Debye–Waller factor and in the Langevin equation.
