|
|
| Line 1: |
Line 1: |
| '''Transient kinetic isotope effects''' (or '''fractionation''') occur when the [[Chemical reaction|reaction]] leading to isotope fractionation does not follow pure [[First order kinetics|first-order kinetics]] and therefore isotopic effects cannot be described with the classical [[equilibrium fractionation]] equations or with steady-state [[Kinetic isotope effect|kinetic fractionation]] equations (also known as the Rayleigh equation).<ref>Mariotti A., J.C. Germon, P. Hubert, P. Kaiser, R. Letolle, A. Tardieux, P. Tardieux, (1981), Experimental determination of nitrogen kinetic isotope fractionation - Some principles - Illustration for the denitrification and nitrification processes, Plant and Soil 62(3), 413-430.</ref> In these instances, the '''General Equations for Biochemical Isotope Kinetics''' ('''GEBIK''') and the '''General Equations for Biochemical Isotope Fractionation''' ('''GEBIF''') can be used. | | If you're pondering that the answer is"The fame and fortune, and the television exposure," then you happen to be in all probability referring to shows like Top rated Chef, Iron Chef, or even perhaps to Hell's Kitchen. The long blade is best for smooth slicing, the curved edge makes it possible for you to rock the knife back and forth generating chopping or mincing a snap, and the rigid shape permits for uniform cuts such as julienne or allumette (the normal French fry reduce). Generally ten inches extended with a serrated edge and a rounded tip, the bread knife comes in handy when slicing bread, cakes, or pastries. Other knives you could possibly want to take into consideration for your arsenal are the boning knife and the carving knife. Set involves: 3-in.<br><br>The entire set is made in Solinge, Germany which is also provided the nick name of the Cutlery Capital of The Globe. A single user said that she has been using this particular set for over a couple of years now and is content with the quality of building and sharpness the knives keep. This is truly an outstanding set of knives for people today who love cooking. As if you happen to be not drooling but, Chef Kurtis Habecker responded boldly in signature original style. Good and easy, this knife looked astounding.<br><br>This 14-piece set stocks a kitchen with most cutlery requirements, and packs all knives in a contemporary black hardwood storage block. The set includes: an 8-inch chef's knife, an eight-inch slicer, an 8-inch bread knife, a 5-inch boning knife, a four-1/2-inch utility knife, a 3-1/two-inch paring knife, and six 4-1/two-inch steak knives, plus a pair of shears. The second season of Top rated Chef started airing in October 2006.<br><br>Filleting knife-This knife is related in size and shape to the boning knife except that it has a versatile as an alternative of a rigid blade. The key to the appropriate use of any knife is making certain it really is sharp. Knife blades are offered an edge on a sharpening stone and maintained among sharpenings by honing with a steel. A chef's knife has a blade 8 to ten inches long and it can reduce through virtually any object. I repeat, do not waste cash on a low cost knife.<br><br>They are serrated on one side Finest Oil To Use For Wood Cutting Boards and beveled on the other, and if they are detachable, the beveled side can be honed with your steel or along with the serrated side sharpened with a knife sharpener developed for both serrated and non-serrated blades! I would suggest, practicing your honing ability on a knife you are not as concerned about ruining. I was offered a set of Worldwide knives final year and I am in knife heaven!<br><br>On the other hand when I was a chef I was really young, and very poor. When it's nice to have a complete set of knives, in a pretty block, along with the matching steak knives, that definitely is not needed. A good simple set must consist of an eight inch French chef's knife, a filet knife, a boning knife, and a paring knife. His added productions are-Morimoto knife & block set, Morimoto specialty beers Imperial Pilsner, Soba Ale, and Black Obi Soba Ale.<br><br>Getting these best rated knife sets is good but it would be amazing recognizing specifically how to use them. When you loved this post and you would want to receive more info relating to [http://Www.Thebestkitchenknivesreviews.com/best-cutting-boards-reviews-guide-and-comparison/ Best Cutting Boards] i implore you to visit our website. The Zwilling J. A. Henckels Comprehensive Book of Knife Skills: The Crucial Guide to Use, Tactics and Care. This spiral bound book has concise step by step directions on a variety of knife methods and security with great is a should have for just about every residence chef. Mastering Knife Abilities: The Essential Guide to the Most Significant Tools in Your Kitchen (with DVD).<br><br>You require to retain your knife sharp at all occasions or else it will turn out to be extra complicated to cut meals, your prep time will improve, and you will need to have to apply far more pressure to finish your chopping job. If items go as planned a chef knife will usually be at hand for an extended period of time. The far more a knife is utilized the much more essential the grip becomes. The manage of most chef knives are now produced of rubber, plastic, or a composite of wood and resin. A further preferred knife is the [http://www.google.Co.uk/search?hl=en&gl=us&tbm=nws&q=Santoku+knife&gs_l=news Santoku knife].<br><br>These significantly less skilled handling a longer knife could possibly come across a santoku knife's shorter blade and lighter all round weight a lot easier to manage. The old planet middleweight champion boxer ended up being arrested for blocking the entrance gates to Downing Street, driving around Parliament square sounding his horn, with a knife in his pocket, and reversing into a delivery van. You start off with the tip of the knife and you gently push it up your stone. |
| | |
| The GEBIK and GEBIF equations are the most generalized approach to describe isotopic effects in any [[Chemical reaction|chemical]], [[Chemical reaction#Catalysis|catalytic reaction]] and [[Chemical reaction#Biochemical reactions|biochemical]] reactions because they can describe isotopic effects in equilibrium reactions, kinetic chemical reactions and kinetic biochemical reactions.<ref>Maggi F. and W. J. Riley, (2010), Mathematical treatment of isotopologue and isotopomer speciation and fractionation in biochemical kinetics, Geochim. Cosmochim. Acta, {{doi|10.1016/j.gca.2009.12.021}}</ref> In the latter two cases, they can describe both stationary and non-stationary fractionation (i.e., variable and inverse fractionation). In general, isotopic effects depend on the number of reactants and on the number of combinations resulting from the number of substitutions in all reactants and products. Describing with accuracy isotopic effects, however, depends also on the specific [[Rate equation|rate law]] used to describe the chemical or biochemical reaction that produces isotopic effects. Normally, regardless of whether a reaction is purely chemical or whether it involves some [[Enzymatic reaction|enzyme]] of biological nature, the equations used to describe isotopic effects base on first-order kinetics. This approach systematically leads to isotopic effects that can be described by means of the Rayleigh equation. In this case, isotopic effects will always be expressed as a constant, hence will not be able to describe isotopic effects in reactions where fractionation and enrichment are variable or inverse during the course of a reaction. Most chemical reactions do not follow first-order kinetics; neither biochemical reactions can normally be described with first-order kinetics. To properly describe isotopic effects in chemical or biochemical reactions, different approaches must be employed such as the use of [[Michaelis–Menten kinetics|Michaelis-Menten]] reaction order (for chemical reactions) or coupled Michaelis-Menten and Monod reaction orders (for biochemical reactions). However, conversely to Michaelis-Menten kinetics, GEBIK and GEBIF equations are solved under the hypothesis of non steady state. This characteristic allows GEBIK and GEBIF to capture ''transient'' isotopic effects.
| |
| | |
| ==Mathematical description of transient kinetic isotope effects==
| |
| The GEBIK and GEBIF equations are introduced here below.
| |
| | |
| ===Notation===
| |
| | |
| The GEBIK and GEBIF equations describe the dynamics of the following state variables
| |
| | |
| :<math> S = </math> substrate concentration
| |
| :<math> P = </math> product concentration
| |
| :<math> E = </math> enzyme concentration
| |
| :<math> C = </math> complex concentration
| |
| :<math> B = </math> biomass concentration
| |
| | |
| Both S and P contain at least one isotopic expression of a tracer atom. For instance, if the carbon element is used as a tracer, both S and P contain at least one C atom, which may appear as <math>^{12}\text{C}</math> and <math>^{13}\text{C}</math>. The isotopic expression within a molecule is
| |
| | |
| :<math>_a^bS</math>
| |
| | |
| where <math>_a</math> is the number of tracer atoms within S, while <math>^b</math> is the number of isotopic substitutions in the same molecule. The condition <math>0 \leq b \leq a </math> must be satisfied. For example, the N<math>_2</math> product in which 1 isotopic substitution occurs (e.g., <math>^{15}\text{N}^{14}\text{N}</math>) will be described by <math>_2^1\text{P}</math>.
| |
| | |
| Substrates and products appear in a chemical reaction with specific stoichiometric coefficients. When chemical reactions comprise combinations of reactants and products with various isotopic expressions, the stoichiometric coefficients are functions of the isotope substitution number. If <math>x_b</math> and <math>y_d</math> are the stoichiometric coefficient for <math>_a^b\text{S}</math> substrate and <math>_c^d\text{P}</math> product, a reaction takes the form
| |
| | |
| :<math>\sum_{b=0}^{a} x_b {_{a}^{b}} \ \text{S} \rightarrow \sum_{d=0}^c y_d {_c^d} \ \text{P}.</math>
| |
| | |
| For example, in the reaction <math>^{14}\text{NO}_3^- +</math> <math>^{15}\text{NO}_3^- \rightarrow^{14}\text{N}^{15}\text{NO}</math>, the notation is <math>{_1^0}\text{S} + {_1^1}\text{S} \rightarrow {_2^1}\text{P}</math> with <math>x_0=x_1=1</math> for both isotopologue reactants of the same substrate with substitution number <math>b=0</math> and <math>b=1</math>, and with <math>y_1=1</math> for <math>_2^1\text{P}</math> and <math>y_0=y_2=0</math> because the reaction does not comprise production of <math>_2^0\text{P} = ^{14}\text{N}_2\text{O}</math> and <math>_2^2\text{P} = ^{15}\text{N}_2\text{O}</math>.
| |
| | |
| For isotopomers, the substitution location is taken into account as <math>_a^b\text{S}^\beta</math> and <math>_a^b\text{S}^\gamma</math> , where <math>\beta</math> and <math>\gamma</math> indicate a different expressions of the same isotopologue <math>_a^b\text{S}</math>. Isotopomers only exist when <math>1 \leq b <a</math> and <math>a \ge 2</math>. The substitution location has to be specifically defined depending on the number of tracer atoms <math>a</math>, number of substitutions <math>b</math>, and molecule structure. For multiatomic molecules that are symmetric with respect to tracer position, there is no need to specify the substitution position when <math>b=1</math>. For example, one substitution of deuterium D = <math>^2\text{H}</math> in the symmetric methane molecule CDH<math>_3</math> does not require the use of the right superscript. In the case that <math>b=2</math>, the substitution location has to be specified, while for CHD<math>_3</math> and CD<math>_4</math> it is not required. For example, two D substitutions in CD<math>_2\text{H}_2</math> can occur in adjacent or non-adjacent locations. Using this notation, the reaction <math>\text{CD}_2\text{H}_2 + 2 \text{O}_2 \rightarrow \text{H}_2\text{O} + \text{D}_2\text{O} + \text{CO}_2,</math> can be written as
| |
| | |
| :<math>{_4^2}\text{S}^{\beta} \rightarrow {_2^0}\text{P} + {_2^2}\text{P}</math>
| |
| | |
| where <math>\beta</math> in <math>{_4^2}\text{S}^{\beta}</math> defines only one of the two methane forms (either with adjacent or non-adjacent D atoms). The location of D in the two isotopologue water molecules produced on the right-hand side of the reaction has not been indicated because D is present in only one water molecule at saturation, and because the water molecule is symmetric. For asymmetric and multiatomic molecules with <math>1\le b<a</math> and <math>a \ge 2</math>, definition of the substitution location is always required. For instance, the isotopomers of the (asymmetric) nitrous oxide molecule <math>\text{N}_2\text{O}</math> are <math>_2^1\text{S}^\beta= {^{15}}\text{N}^{14}\text{NO}</math> and <math>_2^1\text{S}^\gamma= ^{14}\text{N}^{15}\text{NO}</math>.
| |
| | |
| Reactions of asymmetric isotopomers can be written using the partitioning coefficient <math>u</math> as
| |
| | |
| :<math> \sum_{b=0}^{a} \sum_{\beta} x_b \ {_{a}^{b}}\text{S}^{\beta} \rightarrow \sum_{d=0}^c \sum_{\gamma} u_{\gamma} y_d \ {_c^d}\text{P}^{\gamma}, </math>
| |
| | |
| where <math>u_{\gamma}=1</math>. For example, using N isotope tracers, the isotopomer reactions
| |
| | |
| :<math> ^{14}\text{NO}_3^- + ^{15}\text{NO}_3^- \rightarrow {^{14}}\text{N}^{15}\text{NO}, </math>
| |
| :<math> ^{14}\text{NO}_3^- + ^{15}\text{NO}_3^- \rightarrow {^{15}}\text{N}^{14}\text{NO}, </math>
| |
| | |
| can be written as one reaction in which each isotopomer product is multiplied by its partition coefficient as | |
| | |
| :<math>{_1^0}\text{S} + {_1^1}\text{S} \rightarrow u_\beta {_2^1}\text{P}^\beta + u_\gamma {_2^1}\text{P}^\gamma, </math>
| |
| | |
| with <math>u_\gamma=1-u_\beta</math>. More generally, the tracer element does not necessarily occur in only one substrate and one product. If <math>n_S</math> substrates react releasing <math>n_P</math> products, each having an isotopic expression of the tracer element, then the generilized reaction notation is | |
| | |
| :<math>\sum_{j=1}^{n_S} \sum_{b_j=0}^{a_j} \sum_{ \beta_j } x_{b_j} \ {_{a_j}^{b_j}}\text{S}_j^{\beta_j} \rightarrow \sum_{h=1}^{n_P} \sum_{ d_h=0 }^{c_h} \sum_{ \gamma_h } u_{\gamma_h} \ y_{d_h} \ {_{c_h}^{d_h}}\text{P}_h^{\gamma_h}. \qquad \qquad (1) </math>
| |
| | |
| For instance, consider the <math>^{16}\text{O}</math> and <math>^{18}\text{O}</math> tracers in the reaction
| |
| | |
| :<math>\text{CH}_2^{18}\text{O}+^{16}\text{O}_2\rightarrow \text{H}_2^{16}\text{O} +
| |
| \text{C}^{18}\text{O}^{16}\text{O}</math>
| |
| | |
| In this case the reaction can be written as
| |
| | |
| :<math>_1^1\text{S}_1 + _2^0\text{S}_2 \rightarrow {_1^0}\text{P}_1 + {_2^1}\text{P}_2</math>
| |
| | |
| with two substrates and two products without indication of the substitution location because all molecules are symmetric.
| |
| | |
| Biochemical kinetic reactions of type (1) are often catalytic reactions in which one or more substrates, <math>\text{S}_j</math>, bind to an enzyme, E, to form a reversible activated complex, C, which releases one or more
| |
| products, <math>\text{P}_h</math>, and free, unchanged enzyme. These reactions belong to the type of reactions that can be described by [[Michaelis-Menten kinetics|Michaelis-Menten]] kinetics. Using this approach for substrate and product isotopologue and isotopomer expressions, and under the prescribed stoichiometric relationships among them, leads to the general reactions of the Michaelis-Menten type
| |
| | |
| :<math> \sum_{j=1}^{n_S} \sum_{b_{ji}=0}^{a_{ji}} \sum_{ \beta_{ji} } x_{b_{ji}}\ {_{a_j}^{b_{ji}}}\text{S}_j^{\beta_{ji}} + \text{E} \overset{\xrightarrow{\text{k}_{1(i)}}}{
| |
| \xleftarrow[\text{k}_{2(i)}]{} } \text{C}_i \xrightarrow{\text{k}_{3(i)}} \sum_{h=1}^{n_P} \sum_{ d_{hi}=0 }^{c_{hi}} \sum_{ \gamma_{hi} } u_{\gamma_{hi}} y_{d_{hi}} \ {_{c_h}^{d_{hi}}}\text{P}_h^{\gamma_{hi}} + \text{E}, \qquad \qquad (2) </math>
| |
| | |
| with the index <math>i=1,...,m</math>, where <math>m</math> depends on the number of possible atomic combinations among all isotopologues and isotopomers. Here, <math>k_{1(i)}</math>, <math>k_{2(i)}</math>, and <math>k_{3(i)}</math> are the rate constants indexed for each of the ''m'' reactions. | |
| | |
| ==== Example ====
| |
| The reactions
| |
| | |
| :<math> 2 ^{14}\text{NO}_3^- \rightarrow {^{14}}\text{N}_2\text{O}, </math>
| |
| | |
| :<math>^{14}\text{NO}_3^- + {^{15}}\text{NO}_3^- \rightarrow {^{14}}\text{N}^{15}\text{N}\text{O} , </math>
| |
| | |
| :<math>^{14}\text{NO}_3^- + {^{15}}\text{NO}_3^- \rightarrow {^{15}}\text{N}^{14}\text{N}\text{O} , </math>
| |
| | |
| :<math> 2 {^{15}}\text{NO}_3^- \rightarrow {^{15}}\text{N}_2\text{O} , </math>
| |
| | |
| can be written as
| |
| | |
| :<math>2 _1^0\text{S} + \text{E} \overset{\xrightarrow{\text{k}_{1(1)}}}{\xleftarrow[\text{k}_{2(1)}]{}}
| |
| \text{C}_1 \xrightarrow{\text{k}_{3(1)}} {_2^0} \text{P} + \text{E}, </math>
| |
| | |
| :<math> {_1^0}\text{S} + {_1^1}\text{S} + \text{E} \overset{\xrightarrow{\text{k}_{1(2)}}}{ | |
| \xleftarrow[\text{k}_{2(2)}]{} } \text{C}_2 \xrightarrow{\text{k}_{3(2)}} u_\beta {_2^1}\text{P}^\beta + u_\gamma{_2^1}\text{P}^\gamma + \text{E}, </math>
| |
| | |
| :<math> 2 {_1^1}\text{S} + \text{E} \overset{\xrightarrow{\text{k}_{1(3)}}}{ \xleftarrow[\text{k}_{2(3)}]{} } \text{C}_3 \xrightarrow{\text{k}_{3(3)}} {_2^2}\text{P} + \text{E}, </math>
| |
| | |
| ===Isotope mass balance===
| |
| The following isotope mass balances must hold
| |
| | |
| :<math>\sum_{j=1}^{n_S} \sum_{b_j=0}^{a_j} \sum_{ \beta_j } x_{b_j} \ a_j = \sum_{h=1}^{n_P} \sum_{ d_h=0 }^{c_h} \sum_{ \gamma_h } u_{\gamma_h} \ y_{d_h} \ c_h, </math>
| |
| :<math>\sum_{j=1}^{n_S} \sum_{b_j=0}^{a_j} \sum_{ \beta_j } x_{b_j} \ b_j = \sum_{h=1}^{n_P} \sum_{ d_h=0 }^{c_h} \sum_{ \gamma_h } u_{\gamma_h} \ y_{d_h} \ d_{h}. </math>
| |
| | |
| ===General Equations for Biochemical Isotope Kinetics (GEBIK) ===
| |
| | |
| To solve for the concentration of all components appearing in any general biochemical reaction as in (2), the Michaelis-Menten kinetics for an enzymatic reaction are coupled with the Monod kinetics for biomass dynamics. The most general case is to assume that the enzyme concentration is proportional to the biomass concentration and that the reaction is not in quasi-steady state. These hypotheses lead to the following system of equations
| |
| | |
| :<math>\frac{\text{d}[{^{b_j}_{a_j}}S^{\beta_j}_j]}{\text{d}t} = \sum_i
| |
| x_{b_{ji}} [\text{k}_{2(i)} C_i -\text{k}_{1(i)} E \overline{S}_i] \qquad \qquad (3a) </math>
| |
| | |
| :<math>\frac{\text{d}C_i}{\text{d}t} = \text{k}_{1(i)} E
| |
| \overline{S}_i - [\text{k}_{2(i)} + \text{k}_{3(i)}] C_i \qquad \qquad (3b) </math>
| |
| | |
| :<math>\frac{\text{d}[{^{d_h}_{c_h}}P^{\gamma_h}_h]}{\text{d}t} =
| |
| \sum_i u_{\gamma_{hi}} y_{d_{hi}} \text{k}_{3(i)} C_i \qquad \qquad (3c) </math>
| |
| | |
| :<math>\frac{ \text{d}E }{ \text{d}t } = z \frac{\text{d}B}{\text{d}t}
| |
| -\sum_i \frac{\text{d}C_i}{\text{d}t} \qquad \qquad (3d) </math> | |
| | |
| :<math>\frac{ \text{d}B }{ \text{d}t } = Y \sum_h \sum_{d_h}
| |
| \sum_{\gamma_h} \frac{\text{d}[{^{d_h}_{c_h}}P^{\gamma_h}_h]}{\text{d}t} - \mu B \qquad \qquad (3e) </math>.
| |
| | |
| with <math>i=1,...,m</math>, and where <math>\overline{S}_i</math> is the concentration of the most limiting substrate in each reaction ''i'', z is the enzyme yield coefficient, Y is the yield coefficient expressing the biomass gain per unit of released product and <math>\mu</math> is the biomass mortality rate.<ref>Monod J. (1949) The growth of bacterial cultures. Annu. Rev.
| |
| Microbial. 3, 371–394.</ref>
| |
| | |
| === General Equations for Biochemical Isotope Fractionation (GEBIF) ===
| |
| | |
| The isotopic composition of the components in a biochemical system can be defined in different ways depending on the definition of isotopic ratio. Three definitions are described here:
| |
| | |
| ==== Isotopic ratio - definition 1 ====
| |
| | |
| Isotopic ratio relative to each component in the system, each with its isotopic expression, with respect to the concentration of its most abundant
| |
| isotopologue
| |
| | |
| :<math>R^{**}_{S_{b_j,\beta_j}}(t)= \frac{{^{b_j}_{a_j}}S^{\beta_j}_j (t)}{^0_{a_j}S_j(t)} </math>
| |
| | |
| :<math>R^{**}_{P_{d_h,\gamma_h}}(t)= \frac{{^{d_h}_{c_h}}P^{\gamma_h}_h(t)}{^0_{c_h}P_h(t)} </math>
| |
| | |
| ==== Isotopic ratio - definition 2 ====
| |
| | |
| Isotopic ratio relative to the mass of the tracer element in each component;
| |
| | |
| :<math>R^*_{S_j}(t)= \frac{ \displaystyle \sum_{b_j\neq0} \sum_{\beta_j} \frac{b_j q}{ ^{b_j}M_{S_j}} \ {^{b_j}_{a_j}}S^{\beta_j}_j (t)} { \displaystyle \sum_{b_j\neq
| |
| a_j} \sum_{\beta_j} \frac{(a_j-b_j) p}{^{b_j}M_{S_j}} \ {^{b_j}_{a_j}}S^{\beta_j}_j (t)
| |
| } </math>
| |
| | |
| :<math>R^*_{P_h}(t)= \frac{ \displaystyle \sum_{d_h\neq 0} \sum_{\gamma_h} \frac{d_h q}{^{d_h}M_{P_h}} \ {^{d_h}_{c_h}}P^{\gamma_h}_h (t) }{ \displaystyle \sum_{d_h\neq
| |
| c_h} \sum_{\gamma_h} \frac{(c_h-d_h) p }{^{d_h}M_{P_h}} \ {^{d_h}_{c_h}}P^{\gamma_h}_h (t) } </math>
| |
| | |
| where , <math>^{b_j}M_{S_j}</math> and <math>^{d_h}M_{P_h}</math> are the molecular weight of each isotopic expression of the substrate and product.
| |
| | |
| ==== Isotopic ratio - definition 3 ====
| |
| | |
| Isotopic ratio relative to the mass of the tracer element in the accumulated
| |
| substrates and products
| |
| | |
| :<math>R_{S}(t)= \frac{\displaystyle \sum_j \sum_{b_j\neq0} \sum_{\beta_j} \frac{b_j q}{ ^{b_j}M_{S_j}} \ {^{b_j}_{a_j}}S^{\beta_j}_j (t)} { \displaystyle\sum_j
| |
| \sum_{b_j\neq a_j} \sum_{\beta_j} \frac{(a_j-b_j) p}{^{b_j}M_{S_j}} \ {^{b_j}_{a_j}}S^{\beta_j}_j (t) }, </math>
| |
| | |
| :<math>R_{P}(t)= \frac{ \displaystyle\sum_h \sum_{d_h\neq 0} \sum_{\gamma_h} \frac{d_h q}{^{d_h}M_{P_h}} \ {^{d_h}_{c_h}}P^{\gamma_h}_h (t) }{ \displaystyle \sum_h
| |
| \sum_{d_h\neq c_h} \sum_{\gamma_h} \frac{(c_h-d_h) p }{^{d_h}M_{P_h}} \ {^{d_h}_{c_h}}P^{\gamma_h}_h (t) }. </math>
| |
| | |
| ==== Isotopic composition ====
| |
| | |
| Regardless of the definition of the isotopic ratio, the isotopic composition of substrate and product are expressed as
| |
| | |
| :<math>\delta_{S}(t)=\left( \frac{R_{S}(t)}{R_{std}}-1 \right) 1000, \qquad \qquad (4a) </math>
| |
| | |
| :<math>\delta_{P}(t)= \left( \frac{R_{P}(t)}{R_{std}} -1 \right) 1000. \qquad \qquad (4b) </math>
| |
| | |
| where <math>R_{std}</math> is a standard isotopic ration. Here, definition 3 of isotopic ratio has been used, however, any of the three definitions of isotopic ratio can equally be used.
| |
| | |
| ==== Fractionation factor ====
| |
| | |
| The isotopic ratio of the product can be used to define the instantaneous
| |
| isotopic ratio
| |
| | |
| :<math> IR_{P}(t)= \cfrac{ \displaystyle\sum_h \sum_{d_h\neq 0} \sum_{\gamma_h} \cfrac{d_h q}{^{d_h}M_{P_h}} \ \cfrac{\text{d} [{^{d_h}_{c_h}}P^{\gamma_h}_h (t)] }{\text{d} t} }{\displaystyle \sum_h \sum_{d_h\neq c_h} \sum_{\gamma_h} \cfrac{(c_h-d_h) p }{^{d_h}M_{P_h}} \ \cfrac{\text{d} [{^{d_h}_{c_h}}P^{\gamma_h}_h (t)] }{\text{d} t} } \qquad \qquad (5) </math>
| |
| | |
| and the time-dependent fractionation factor
| |
| | |
| :<math> \alpha(t)=\frac{IR_{P}(t)}{R_S(t)} \qquad \qquad (6) </math> | |
| | |
| ==== Isotopic enrichment ====
| |
| | |
| The time-dependent isotopic enrichment is simply defined as | |
| | |
| :<math> \epsilon(t)=1- \alpha(t) \qquad \qquad (7) </math>
| |
| | |
| == Simplified forms of GEBIK and GEBIF ==
| |
| | |
| Under specific assumptions, the GEBIK and GEBIF equations become equivalent to the equation for steady-state kinetic isotope fractionation in both chemical and biochemical reactions. Here two mathematical treatments are proposed: (i) under '''biomass-free''' and '''enzyme-invariant''' (BFEI) hypothesis and (ii) under '''quasi-steady-state''' (QSS) hypothesis.
| |
| | |
| === BFEI hypothesis ===
| |
| In instances where the biomass and enzyme concentrations are not appreciably changing in time, we can assume that biomass dynamics is negligible and that the total enzyme concentration is constant, and the GEBIK equations become
| |
| | |
| :<math>\frac{\text{d}[{^{b_j}_{a_j}}S^{\beta_j}_j]}{\text{d}t} = \sum_i x_{b_{ji}} [\text{k}_{2(i)} C_i -\text{k}_{1(i)} E \overline{S}_i] \qquad \qquad (8a) </math>
| |
| | |
| :<math>\frac{\text{d}C_i}{\text{d}t} = \text{k}_{1(i)} E \overline{S}_i - [\text{k}_{2(i)} + \text{k}_{3(i)}] C_i \qquad \qquad (8b) </math>
| |
| | |
| :<math> \frac{\text{d}[{^{d_h}_{c_h}}P^{\gamma_h}_h]}{\text{d}t} = \sum_i u_{\gamma_{hi}} y_{d_{hi}} \text{k}_{3(i)} C_i \qquad \qquad (8c) </math>
| |
| | |
| :<math>\frac{ \text{d}E }{ \text{d}t } = - \sum_i \frac{\text{d}C_i}{\text{d}t} \qquad \qquad (8d) </math>
| |
| | |
| Eqs. (4) for isotopic compositions, Eq. (6) for the fractionation factor and Eq. (7) for the enrichment factor equally applies to the GEBIK equations under the BFEI hypothesis.
| |
| | |
| === QSS hypothesis ===
| |
| If the quasi-steady-state hypothesis is assumed in addition to BFEI hypothesis, then the complex concentration can be assumed to be in a stationary (steady) state according to the Briggs-[[J. B. S. Haldane#Population genetics and the Briggs–Haldane equation|Haldane]] hypothesis, and the GEBIK equations become
| |
| | |
| :<math> \frac{ \text{d} [{_{a_j}^{b_{j}}}S_j^{\beta_{j}} ] }{ \text{d}t } \simeq - \sum_{i=1}^m \frac{ x_{b_{ji}} \text{k}_{3(i)} E_0 \overline{S}_i }{ \overline{S}_i + K_i \left( 1+ \displaystyle\sum_{p\neq i} \dfrac{\overline{S}_p}{K_p} \right) } \qquad \qquad (9a) </math>
| |
| | |
| :<math> \frac{\text{d} [{^{d_h}_{c_h}}P^{\gamma_h}_h ]}{\text{d}t} \simeq \sum_{i=1}^m \frac{ u_{\gamma_{hi}} y_{d_{hi}} \text{k}_{3(i)} E_0 \overline{S}_i }{ \overline{S}_i + K_i \left(
| |
| 1+ \displaystyle\sum_{p\neq i} \dfrac{\overline{S}_p}{K_p} \right) } \qquad \qquad (9b) </math>
| |
| | |
| which are written in a form similar to the classical Micaelis-Menten equations for any substrate and product. Here, the equations also show that the various isotopologue and isotopomer substrates appear as competing species. Eqs. (4) for isotopic compositions, Eq. (6) for the fractionation factor and Eq. (7) for the enrichment factor equally applies to the GEBIK equations under the BFEI and QSS hypothesis.
| |
| | |
| == Example of application of GEBIK and GEBIF ==
| |
| | |
| An example is shown where GEBIK and GEBIF equations are used to describe the isotopic reactions of N<math>_2</math>O consumption into N<math>_2</math> according to the simultaneous set of reactions
| |
| | |
| :<math> ^{14}\text{N}_2\text{O} \rightarrow {^{14}}\text{N}_2, </math>
| |
| | |
| :<math> {^{14}}\text{N}^{15}\text{N}\text{O} \rightarrow {^{14}}\text{N}{^{15}}\text{N}, </math>
| |
| | |
| :<math> {^{15}}\text{N}^{14}\text{N}\text{O} \rightarrow {^{14}}\text{N}{^{15}}\text{N}, </math>
| |
| | |
| These can be rewritten using the notation introduced before as.
| |
| | |
| :<math>_2^0\text{S} + \text{E} \underset{\text{k}_{2(1)}}{\overset{\text{k}_{1(1)}}{\rightleftarrows}}
| |
| \text{C}_1 \overset{\text{k}_{3(1)}}{\rightarrow} {_2^0} \text{P} + \text{E}, </math>
| |
| | |
| :<math> {_2^1}\text{S}^\beta + \text{E} \underset{\text{k}_{2(2)}}{\overset{\text{k}_{1(2)}}{\rightleftarrows}} \text{C}_2 \overset{\text{k}_{3(2)}}{\rightarrow} {_2^1}\text{P} + \text{E}, </math>
| |
| | |
| :<math> {_2^1}\text{S}^\gamma + \text{E} \underset{\text{k}_{2(3)}}{\overset{\text{k}_{1(3)}}{\rightleftarrows}} \text{C}_3 \overset{\text{k}_{3(3)}}{\rightarrow} {_2^1}\text{P} + \text{E}, </math>
| |
| | |
| The substrate <math>_2^2\text{S} = ^{15}\text{N}_2\text{O}</math> has not been included due to its scarcity. In addition, we have not specified the isotopic substitution in the N<math>_2</math> product of the second and third reactions because N<math>_2</math> is symmetric. Assuming that the second and third reactions have identical reaction rates <math>(k_{1(3)}\equiv k_{1(2)}</math>, <math>k_{2(3)}\equiv k_{2(2)}</math>, and <math>k_{3(3)}\equiv k_{3(2)})</math>, the full GEBIK and GEBIF equations are
| |
| | |
| :<math>\frac{\text{d} [{^0_2}S]}{\text{d}t} = \text{k}_{2(1)} C_1 -\text{k}_{1(1)} {^0_2}S E </math>
| |
| | |
| :<math>\frac{\text{d} [{^1_2}S^\beta]}{\text{d}t} = \text{k}_{2(2)} C_2 -\text{k}_{1(2)} {^1_2}S^\beta E</math> | |
| | |
| :<math>\frac{\text{d} [{^1_2}S^\gamma]}{\text{d}t}= \text{k}_{2(2)} C_3 -\text{k}_{1(2)} {^1_2}S^\gamma E</math>
| |
| | |
| :<math>\frac{\text{d}C_1}{\text{d}t} = \text{k}_{1(1)} {^0_2}S E - (\text{k}_{2(1)} + \text{k}_{3(1)}) C_1</math>
| |
| | |
| :<math>\frac{\text{d}C_2}{\text{d}t}= \text{k}_{1(2)} {^1_2}S^\beta E - (\text{k}_{2(2)}+ \text{k}_{3(2)}) C_2</math>
| |
| | |
| :<math>\frac{\text{d}C_3}{\text{d}t}= \text{k}_{1(2)} {^1_2}S^\gamma E - (\text{k}_{2(2)}+ \text{k}_{3(2)}) C_3</math>
| |
| | |
| :<math>\frac{\text{d} [{_2^0}P]}{\text{d}t} = \text{k}_{3(1)} C_1</math>
| |
| | |
| :<math>\frac{\text{d} [{_2^1}P]}{\text{d}t} = \text{k}_{3(2)} (C_2 + C_3)</math>
| |
| | |
| :<math>\frac{ \text{d}E }{ \text{d}t } = z \frac{\text{d}B}{\text{d}t} - \frac{\text{d}C_1}{\text{d}t} - \frac{\text{d}C_2}{\text{d}t} - \frac{\text{d}C_3}{\text{d}t}</math>
| |
| | |
| :<math>\frac{ \text{d}B }{ \text{d}t } = Y \left( \frac{\text{d} [{_2^0}P]}{\text{d}t} + \frac{\text{d} [{_2^1}P]}{\text{d}t} \right) - \mu B</math>
| |
| | |
| :<math> R_P(t) = \frac{ 15 \ {_2^1}P(t) }{ 14 \ {_2^1}P(t) + 29 \ {_2^0}P(t) }</math>
| |
| | |
| :<math> IR_{P}(t) = \dfrac{ 15 \ (C_2 + C_3) \text{k}_{3(2)} } { 29 \ C_1 \text{k}_{3(1)} + 14 \ (C_2 + C_3) \text{k}_{3(2)} }, </math>
| |
| | |
| :<math> R_{S}(t) = \frac{ 165 \ {_2^1}S }{ 154 \ {_2^1}S + 315 \ {_2^0}S }</math>
| |
| | |
| :<math> \alpha(t) = \frac{ 7 \ (C_2 + C_3) \text{k}_{3(2)} [45 \ {_2^0}S + 22 \ {_2^1}S] } { 11 \ [29 \ C_1 \text{k}_{3(1)} + 14 \ (C_2 + C_3) \text{k}_{3(2)} ] \ {_2^1}S } </math>
| |
| | |
| == Example of application of GEBIK and GEBIF under BFEI and QSS hypotheses==
| |
| | |
| The same reaction can be described with the GEBIK and GEBIF equations under the BFEI and QSS approximations as | |
| | |
| :<math>\frac{\text{d}[{^0_2}S]}{\text{d}t} \simeq - \frac{\text{k}_{3(1)} E_0 {^0_2}S }{ ^0_2S + K_1 \left( 1+ \dfrac{{^1_2}S^\beta }{ K_2} + \dfrac{ {^1_2}S^\gamma }{ K_2} \right)} </math>
| |
| | |
| :<math>\frac{\text{d}[{^1_2}S^\beta]}{\text{d}t} \simeq - \frac{ \text{k}_{3(2)} E_0 {^1_2}S^\beta }{ ^1_2S^\beta + K_2 \left( 1+ \dfrac{ {^0_2}S }{ K_1} + \dfrac{ {^1_2}S^\gamma }{ K_2} \right) }</math>
| |
| | |
| :<math>\frac{\text{d}[{^1_2}S^\gamma]}{\text{d}t} \simeq - \frac{ \text{k}_{3(2)} E_0 {^1_2}S^\gamma }{ ^1_2S^\gamma + K_2 \left( 1+ \dfrac{ {^0_2}S }{K_1} + \dfrac{ {^1_2}S^\beta }{ K_2} \right) }</math>
| |
| | |
| :<math> \frac{\text{d } {_2^0}P}{\text{d}t} = - \frac{\text{d}[{^0_2}S]}{\text{d}t} </math>
| |
| | |
| :<math> \frac{\text{d } {_2^1}P}{\text{d}t} = - \frac{\text{d}[{^1_2}S^\beta]}{\text{d}t} - \frac{\text{d}[{^1_2}S^\gamma]}{\text{d}t}</math>
| |
| | |
| :<math> R_P(t) = \frac{ 15 {_2^1}P }{ 14 {_2^1}P + 29 \ {_2^0}P }</math>
| |
| | |
| :<math> IR_{P}(t) = \dfrac{ 15 K_1 \text{k}_{3(2)} {_2^1}S }{ 29 K_2 \text{k}_{3(1)} {_2^0}S + 14 K_1 \text{k}_{3(2)} {_2^1}S }</math>
| |
| | |
| :<math> R_{S}(t) = \frac{ 465 \ {_2^1}S }{ 14 [63 \ {_2^0}S + 31 \ {_2^1}S] }</math>
| |
| | |
| :<math> \alpha(t) = \frac{ 14 K_1 \text{k}_{3(2)} [63 \ {_2^0}S + 31 \ {_2^1}S] }{ 31 [29 K_2 \text{k}_{3(1)} \ {_2^0}S + 14 K_1 \text{k}_{3(2)} \ {_2^0}S ] }</math>
| |
| | |
| where K<math>_3</math> has been substituted with K<math>_2</math> because the rate
| |
| constants in the third reaction have been assumed to equal those
| |
| of the second reaction. | |
| | |
| ==See also==
| |
| *[[Equilibrium fractionation]]
| |
| *[[Isotope electrochemistry]]
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| [[Category:Chemical kinetics]]
| |
| [[Category:Isotope separation]]
| |
If you're pondering that the answer is"The fame and fortune, and the television exposure," then you happen to be in all probability referring to shows like Top rated Chef, Iron Chef, or even perhaps to Hell's Kitchen. The long blade is best for smooth slicing, the curved edge makes it possible for you to rock the knife back and forth generating chopping or mincing a snap, and the rigid shape permits for uniform cuts such as julienne or allumette (the normal French fry reduce). Generally ten inches extended with a serrated edge and a rounded tip, the bread knife comes in handy when slicing bread, cakes, or pastries. Other knives you could possibly want to take into consideration for your arsenal are the boning knife and the carving knife. Set involves: 3-in.
The entire set is made in Solinge, Germany which is also provided the nick name of the Cutlery Capital of The Globe. A single user said that she has been using this particular set for over a couple of years now and is content with the quality of building and sharpness the knives keep. This is truly an outstanding set of knives for people today who love cooking. As if you happen to be not drooling but, Chef Kurtis Habecker responded boldly in signature original style. Good and easy, this knife looked astounding.
This 14-piece set stocks a kitchen with most cutlery requirements, and packs all knives in a contemporary black hardwood storage block. The set includes: an 8-inch chef's knife, an eight-inch slicer, an 8-inch bread knife, a 5-inch boning knife, a four-1/2-inch utility knife, a 3-1/two-inch paring knife, and six 4-1/two-inch steak knives, plus a pair of shears. The second season of Top rated Chef started airing in October 2006.
Filleting knife-This knife is related in size and shape to the boning knife except that it has a versatile as an alternative of a rigid blade. The key to the appropriate use of any knife is making certain it really is sharp. Knife blades are offered an edge on a sharpening stone and maintained among sharpenings by honing with a steel. A chef's knife has a blade 8 to ten inches long and it can reduce through virtually any object. I repeat, do not waste cash on a low cost knife.
They are serrated on one side Finest Oil To Use For Wood Cutting Boards and beveled on the other, and if they are detachable, the beveled side can be honed with your steel or along with the serrated side sharpened with a knife sharpener developed for both serrated and non-serrated blades! I would suggest, practicing your honing ability on a knife you are not as concerned about ruining. I was offered a set of Worldwide knives final year and I am in knife heaven!
On the other hand when I was a chef I was really young, and very poor. When it's nice to have a complete set of knives, in a pretty block, along with the matching steak knives, that definitely is not needed. A good simple set must consist of an eight inch French chef's knife, a filet knife, a boning knife, and a paring knife. His added productions are-Morimoto knife & block set, Morimoto specialty beers Imperial Pilsner, Soba Ale, and Black Obi Soba Ale.
Getting these best rated knife sets is good but it would be amazing recognizing specifically how to use them. When you loved this post and you would want to receive more info relating to Best Cutting Boards i implore you to visit our website. The Zwilling J. A. Henckels Comprehensive Book of Knife Skills: The Crucial Guide to Use, Tactics and Care. This spiral bound book has concise step by step directions on a variety of knife methods and security with great is a should have for just about every residence chef. Mastering Knife Abilities: The Essential Guide to the Most Significant Tools in Your Kitchen (with DVD).
You require to retain your knife sharp at all occasions or else it will turn out to be extra complicated to cut meals, your prep time will improve, and you will need to have to apply far more pressure to finish your chopping job. If items go as planned a chef knife will usually be at hand for an extended period of time. The far more a knife is utilized the much more essential the grip becomes. The manage of most chef knives are now produced of rubber, plastic, or a composite of wood and resin. A further preferred knife is the Santoku knife.
These significantly less skilled handling a longer knife could possibly come across a santoku knife's shorter blade and lighter all round weight a lot easier to manage. The old planet middleweight champion boxer ended up being arrested for blocking the entrance gates to Downing Street, driving around Parliament square sounding his horn, with a knife in his pocket, and reversing into a delivery van. You start off with the tip of the knife and you gently push it up your stone.