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Biochemistry
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Biochemistry, sometimes called biological chemistry, is the study of chemical processes within and relating to living organisms. By controlling information flow through biochemical signaling and the flow of energy through metabolism. Biochemistry is closely related to biology, the study of the molecular mechanisms by which genetic information encoded in DNA is able to result in the processes of life. Depending on the definition of the terms used, molecular biology can be thought of as a branch of biochemistry, or biochemistry as a tool with which to investigate. The chemistry of the cell depends on the reactions of smaller molecules. These can be inorganic, for water and metal ions, or organic, for example the amino acids. The mechanisms by which cells harness energy from their environment via chemical reactions are known as metabolism, the findings of biochemistry are applied primarily in medicine, nutrition, and agriculture. In medicine, biochemists investigate the causes and cures of diseases, in nutrition, they study how to maintain health and study the effects of nutritional deficiencies. In agriculture, biochemists investigate soil and fertilizers, and try to discover ways to improve crop cultivation, crop storage and pest control. However, biochemistry as a scientific discipline has its beginning sometime in the 19th century, or a little earlier. Gowland Hopkins on enzymes and the nature of biochemistry. The term biochemistry itself is derived from a combination of biology, the German chemist Carl Neuberg however is often cited to have coined the word in 1903, while some credited it to Franz Hofmeister. Then, in 1828, Friedrich Wöhler published a paper on the synthesis of urea and these techniques allowed for the discovery and detailed analysis of many molecules and metabolic pathways of the cell, such as glycolysis and the Krebs cycle. Another significant historic event in biochemistry is the discovery of the gene and this part of biochemistry is often called molecular biology. In the 1950s, James D. Watson, Francis Crick, Rosalind Franklin, in 1958, George Beadle and Edward Tatum received the Nobel Prize for work in fungi showing that one gene produces one enzyme. In 1988, Colin Pitchfork was the first person convicted of murder with DNA evidence, mello received the 2006 Nobel Prize for discovering the role of RNA interference, in the silencing of gene expression. Around two dozen of the 92 naturally occurring elements are essential to various kinds of biological life. Most rare elements on Earth are not needed by life, while a few common ones are not used, most organisms share element needs, but there are a few differences between plants and animals
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Enzyme kinetics
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Enzyme kinetics is the study of the chemical reactions that are catalysed by enzymes. In enzyme kinetics, the rate is measured and the effects of varying the conditions of the reaction are investigated. Enzymes are usually protein molecules that manipulate other molecules — the enzymes substrates, kinetic studies on enzymes that only bind one substrate, such as triosephosphate isomerase, aim to measure the affinity with which the enzyme binds this substrate and the turnover rate. Some other examples of enzymes are phosphofructokinase and hexokinase, both of which are important for cellular respiration, when enzymes bind multiple substrates, such as dihydrofolate reductase, enzyme kinetics can also show the sequence in which these substrates bind and the sequence in which products are released. An example of enzymes that bind a single substrate and release multiple products are proteases, others join two substrates together, such as DNA polymerase linking a nucleotide to DNA. Although these mechanisms are often a series of steps, there is typically one rate-determining step that determines the overall kinetics. This rate-determining step may be a reaction or a conformational change of the enzyme or substrates. Knowledge of the structure is helpful in interpreting kinetic data. For example, the structure can suggest how substrates and products bind during catalysis, what changes occur during the reaction, not all biological catalysts are protein enzymes, RNA-based catalysts such as ribozymes and ribosomes are essential to many cellular functions, such as RNA splicing and translation. The main difference between ribozymes and enzymes is that RNA catalysts are composed of nucleotides, whereas enzymes are composed of amino acids, ribozymes also perform a more limited set of reactions, although their reaction mechanisms and kinetics can be analysed and classified by the same methods. The reaction catalysed by an enzyme uses exactly the same reactants, like other catalysts, enzymes do not alter the position of equilibrium between substrates and products. However, unlike uncatalysed chemical reactions, enzyme-catalysed reactions display saturation kinetics, the substrate concentration midway between these two limiting cases is denoted by KM. The two most important kinetic properties of an enzyme are how quickly the enzyme becomes saturated with a substrate. Knowing these properties suggests what an enzyme might do in the cell, Enzyme assays are laboratory procedures that measure the rate of enzyme reactions. Because enzymes are not consumed by the reactions they catalyse, enzyme assays usually follow changes in the concentration of substrates or products to measure the rate of reaction. There are many methods of measurement, spectrophotometric assays are most convenient since they allow the rate of the reaction to be measured continuously. Although radiometric assays require the removal and counting of samples they are extremely sensitive. An analogous approach is to use mass spectrometry to monitor the incorporation or release of stable isotopes as substrate is converted into product, the most sensitive enzyme assays use lasers focused through a microscope to observe changes in single enzyme molecules as they catalyse their reactions
3.
Reaction rate
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The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how quickly or slowly a reaction takes place. Chemical kinetics is the part of chemistry that studies reaction rates. The concepts of chemical kinetics are applied in many disciplines, such as engineering, enzymology. Consider a typical reaction, a A + b B → p P + q Q The lowercase letters represent stoichiometric coefficients, while the capital letters represent the reactants. Reaction rate usually has the units of mol L−1 s−1, the rate of a reaction is always positive. A negative sign is present to indicate that the reactant concentration is decreasing. )The IUPAC recommends that the unit of time should always be the second. The rate of reaction differs from the rate of increase of concentration of a product P by a constant factor and for a reactant A by minus the reciprocal of the stoichiometric number. The stoichiometric numbers are included so that the rate is independent of which reactant or product species is chosen for measurement. For example, if a =1 and b =3 then B is consumed three times more rapidly than A, but v = -d/dt = -d/dt is uniquely defined. The above definition is valid for a single reaction, in a closed system of constant volume. If water is added to a pot containing salty water, the concentration of salt decreases, although there is no chemical reaction. When applied to the system at constant volume considered previously, this equation reduces to, r = d d t. Here N0 is the Avogadro constant, for a single reaction in a closed system of varying volume the so-called rate of conversion can be used, in order to avoid handling concentrations. It is defined as the derivative of the extent of reaction with respect to time, also V is the volume of reaction and Ci is the concentration of substance i. When side products or reaction intermediates are formed, the IUPAC recommends the use of the rate of appearance and rate of disappearance for products and reactants. Reaction rates may also be defined on a basis that is not the volume of the reactor, when a catalyst is used the reaction rate may be stated on a catalyst weight or surface area basis. If the basis is a specific catalyst site that may be counted by a specified method. The nature of the reaction, Some reactions are faster than others
4.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
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Concentration
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In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished, mass concentration, molar concentration, number concentration, the term concentration can be applied to any kind of chemical mixture, but most frequently it refers to solutes and solvents in solutions. The molar concentration has variants such as concentration and osmotic concentration. To concentrate a solution, one must add more solute, or reduce the amount of solvent, by contrast, to dilute a solution, one must add more solvent, or reduce the amount of solute. Unless two substances are fully miscible there exists a concentration at which no further solute will dissolve in a solution, at this point, the solution is said to be saturated. If additional solute is added to a solution, it will not dissolve, except in certain circumstances. Instead, phase separation will occur, leading to coexisting phases, the point of saturation depends on many variables such as ambient temperature and the precise chemical nature of the solvent and solute. Concentrations are often called levels, reflecting the mental schema of levels on the axis of a graph. There are four quantities that describe concentration, The mass concentration ρ i is defined as the mass of a constituent m i divided by the volume of the mixture V, ρ i = m i V. The molar concentration c i is defined as the amount of a constituent n i divided by the volume of the mixture V, c i = n i V, however, more commonly the unit mol/L is used. The number concentration C i is defined as the number of entities of a constituent N i in a divided by the volume of the mixture V, C i = N i V. The volume concentration ϕ i is defined as the volume of a constituent V i divided by the volume of the mixture V, ϕ i = V i V. Being dimensionless, it is expressed as a number, e. g.0.18 or 18%, several other quantities can be used to describe the composition of a mixture. Note that these should not be called concentrations, normality is defined as the molar concentration c i divided by an equivalence factor f e q. Since the definition of the equivalence factor depends on context, IUPAC, the SI unit for molality is mol/kg. The mole fraction x i is defined as the amount of a constituent n i divided by the amount of all constituents in a mixture n t o t, x i = n i n t o t. However, the deprecated parts-per notation is used to describe small mole fractions. The mole ratio r i is defined as the amount of a constituent n i divided by the amount of all other constituents in a mixture
6.
Nonlinear regression
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The data are fitted by a method of successive approximations. The data consist of independent variables, x, and their associated observed dependent variables. Each y is modeled as a variable with a mean given by a nonlinear function f. Systematic error may be present but its treatment is outside the scope of regression analysis, if the independent variables are not error-free, this is an errors-in-variables model, also outside this scope. This function is nonlinear because it cannot be expressed as a combination of the two β s. Other examples of nonlinear functions include exponential functions, logarithmic functions, trigonometric functions, power functions, Gaussian function, some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear. When so transformed, standard linear regression can be performed but must be applied with caution, see Linearization, below, for more details. In general, there is no closed-form expression for the best-fitting parameters, usually numerical optimization algorithms are applied to determine the best-fitting parameters. Again in contrast to linear regression, there may be many local minima of the function to be optimized, in practice, estimated values of the parameters are used, in conjunction with the optimization algorithm, to attempt to find the global minimum of a sum of squares. For details concerning nonlinear data modeling see least squares and non-linear least squares, the assumption underlying this procedure is that the model can be approximated by a linear function. F ≈ f 0 + ∑ j J i j β j where J i j = ∂ f ∂ β j and it follows from this that the least squares estimators are given by β ^ ≈ −1 J T y. The nonlinear regression statistics are computed and used as in linear regression statistics, the linear approximation introduces bias into the statistics. Therefore more caution than usual is required in interpreting statistics derived from a nonlinear model, the best-fit curve is often assumed to be that which minimizes the sum of squared residuals. This is the least squares approach, however, in cases where the dependent variable does not have constant variance, a sum of weighted squared residuals may be minimized, see weighted least squares. Each weight should ideally be equal to the reciprocal of the variance of the observation, some nonlinear regression problems can be moved to a linear domain by a suitable transformation of the model formulation. For example, consider the nonlinear regression problem y = a e b x U with parameters a and b, however, use of a nonlinear transformation requires caution. The influences of the values will change, as will the error structure of the model. These may not be desired effects, for Michaelis–Menten kinetics, the linear Lineweaver–Burk plot 1 v =1 V max + K m V max of 1/v against 1/ has been much used
![{\displaystyle v=-K_{M}{v \over [S]}+V_{\max }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e64e5ff5aa6bad2028d9eb6fe48e38eff8f914)
![{\displaystyle v={{V_{\max }{}[S]} \over {K_{M}+[S]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dab71ee907aff1a65789538b8db961da128ede5)
![{\displaystyle {V_{\max } \over v}={{V_{\max }{}(K_{M}+[S])} \over {V_{\max }{}[S]}}={{K_{M}+[S]} \over {[S]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d7ad3bf670e453f892ecd8ca40da1f61d42c2cb)
![{\displaystyle V_{\max }={{{vK_{M}} \over {[S]}}+{{v[S]} \over {[S]}}}={{vK_{M}} \over {[S]}}+v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13153fe62a963ee0e7a0074343538609d1b428df)
![{\displaystyle v=-K_{M}{v \over {[S]}}+V_{\max }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af6684193dd23ac4e924d699b22a913a6da0ef4a)
. PMID 1201002.