In induced fit, a substrate binds to an enzyme and the resulting enzyme-substrate complex subsequently changes its protein structure. This adjustment can position interacting groups at the active site for catalysis. The mechanistic claim concerns order: binding precedes the relevant change. In conformational selection, the unbound enzyme already visits an alternative protein structure, which the substrate preferentially binds. Both processes can occur in one system. Let and denote two structures of the same enzyme, and a substrate. The limiting routes areThe star labels a structure, not a different protein primary structure, and does not by itself mean catalytic activity. Each reversible arrow abbreviates a step, not the whole catalytic cycle. See Gordon G. Hammes, Yu-Chu Chang and Terrence G. Oas, Conformational selection or induced fit: a flux description of reaction mechanism, PNAS 106 (2009), 13737–13741.
A rigid lock-and-key cartoon usefully suggests selective substrate recognition, but it suppresses movement of the protein and chemical changes at the active site. Replacing it with two static drawings, “open” and “closed”, still does not establish induced fit: both routes above can end at . Also, if complete closure prevents a substrate entering, the pre-existing structure selected in conformational selection might be partly open and undergo further induced fit after binding. The usefulness of the distinction is that it turns an attractive shape story into questions about enzyme kinetics. Relative traffic through the routes depends on populations, rate constants and free substrate concentration, rather than affinity alone. See Hammes, Chang and Oas, 2009: coupled pathways, their concentrations and the limitations of endpoint structures.
Induced fit and conformational selection: two orders, one bound endpoint
. Original reaction-order schematic and an analytical first-event model, not a hexokinase structure or measured data. In the upper square, horizontal steps bind or release a substrate; vertical steps change protein structure. In the lower plot, an initially unbound can bind with rate or change structure with rate , for chosen and . The plotted probability concerns only the first event from , not the fraction completing either full route. Subsequent binding, reversals and catalysis are not modelled by that curve. Details.
For this analytical illustration, hold the free substrate concentration constant and assume independent waiting times with exponential distributions of constant rates. The law of mass action gives the binding rate ; the competing change has rate . The probability density function for binding to happen first at time is , because neither event has happened earlier. Its integral isWith the stated invented rate constants, gives , respectively. Raising free substrate concentration makes binding more likely to win this initial competition. This probability is deliberately narrower than a full-route fraction: returns to , later changes and the binding properties of would all matter for the latter. The illustration explains why timing and concentration must accompany structural evidence, without assigning a binding mechanism to a real enzyme.
A motivating experiment tested the crystal interpretation in solution. McDonald, Steitz and Engelman prepared solutions of monomeric yeast hexokinase B and compared the unbound preparation with sugar-bound preparations. Their apparatus measured small-angle scattering of X-rays; the spatial extent inferred from that readout was the radius of gyration, rather than a catalytic reaction rate. Binding glucose reduced the radius of gyration by ; binding the phosphorylated glucose product reduced it by , as reported by the authors. The glucose change agreed with the reduction calculated from the crystal coordinates. See Robert C. McDonald, Thomas A. Steitz and Donald M. Engelman, Yeast hexokinase in solution exhibits a large conformational change upon binding glucose or glucose 6-phosphate, Biochemistry 18 (1979), 338–342: primary abstract.
The comparison without added substrate supplies a baseline within the same form of hexokinase; the second bound preparation tests another binding condition. Together with the crystal comparison, solution compaction supports closure beyond crystal packing. It still gives an average spatial readout, not a movie of individual binding events, and product-associated compaction does not demonstrate ongoing catalysis. Neither experiment alone proves that binding precedes movement. To distinguish induced fit from conformational selection, structural comparisons must be combined with measurements sensitive to timing and with a model of the relevant populations and enzyme kinetics.
For a separated acoustic potential with transverse Neumann eigenvalue , integration by parts gives . The time-averaged kinetic energy is proportional to , while compression energy is proportional to . The dispersion relation proves equality. At a ray speed below , nondegenerate stationary phase gives dispersive amplitudes of order ; the zero transverse mode instead preserves an undispersed pulse on the sound-speed ray.
For rigid plates , , an even incident plane wave travelling toward the opening can be scattered using Wiener-Hopf factorization. With time dependence and transform , take on the outgoing sheet. The common normal derivative on is supported on ; its transform is . The interior transform is , while the exterior transform is . These follow from continuity of normal velocity and the half-line pressure jump, followed by pole subtraction in a Wiener-Hopf equation and a finite-energy edge condition.
For the even acoustic waveguide mode , set and . At leading independent-edge order for large , the two incident travelling constituents each have amplitude at the relevant wall endpoint. Reflecting the upper-edge coordinates reverses the edge-wave sign. Their path phases are . Thus Sommerfeld half-plane diffraction gives the displayed sum, where . For , the apparent poles at are removable in the combined expression, with limit . Single-edge nonuniform far-field poles are not physical infinities. This combined far-field limit assumes a radiation distance beyond the aperture's near field and neglects further edge interactions; it is not uniform at duct cutoff.
In a stationary, horizontally homogeneous wall boundary layer, Hamiltonian ray-tracing equations conserve and . If increases away from the wall, downstream-oriented waves have decreasing positive intrinsic frequency and decreasing . At , the transverse wavenumber vanishes and the acoustic ray turns. For , a downstream launch angle above the wall can reach a uniform stream of Mach number only if . This uses the positive acoustic branch, constant sound speed and no intervening critical layer; a turning point itself requires a uniform local approximation rather than the ordinary WKB method.
The acid dissociation constant quantifies the chemical equilibrium for loss of one proton from a chemical acid. In water, write the actual proton transfer as . Its full dimensionless equilibrium constant isThe usual aqueous acid dissociation constant uses the solvent convention with water thermodynamic activity approximated by one:Here hydrated hydrogen ions are understood; is shorthand, not an isolated bare proton in the chemical solution. All thermodynamic activities and are dimensionless quantities, using consistent standard states. If water thermodynamic activity differs appreciably from one, retain it in the full equilibrium constant instead of silently mixing conventions.
With activity coefficients and reference concentration , put for each solute. ThenThe last expression is the dilute ideal concentration approximation, not the definition in a nonideal chemical solution. Omitting instead produces a quotient with units of concentration; its numerical value depends on the chosen units. Do not take a logarithm of that dimensional quotient. Conditional concentration quotients can change with ionic strength even at fixed temperature, whereas the thermodynamic acid dissociation constant retains the activity coefficient corrections. The acid dissociation constant entry distinguishes these conventions.
At the same temperature, solvent and standard states, a larger acid dissociation constant, equivalently a smaller pKa, indicates a stronger tendency to donate a proton. It does not specify an analytical concentration, a pH, or a reaction rate. Composition and the weak acid mass and charge balances are needed to predict pH; chemical kinetics addresses how quickly the chemical equilibrium is approached.
A weak acid establishes a chemical equilibrium between its proton-donating form and its conjugate base in water. For a neutral monoprotic acid , the aqueous chemical equation isWater accepts a proton to become hydronium; is the conjugate base. The double arrow denotes reversible exchange at chemical equilibrium, not equal populations. The acid dissociation constant and composition determine the hydronium concentration; substituting total chemical acid concentration into the pH formula is generally incorrect.
“Weak” describes the tendency to transfer a proton in a specified solvent, rather than how much chemical acid was dissolved. A weak acid can be concentrated, and a strong acid can be dilute. Even the same weak acid can have a large weak acid dissociation fraction after sufficient dilution, while its acid dissociation constant stays fixed at the same temperature and standard state conventions.[ref]
14.3 Relative Strengths of Acids and Bases. Chemistry 2e by Paul Flowers and others; §14.3, acid ionization and concentration-dependent percent ionization
In water, a strong acid transfers its available proton essentially completely to water under the dilute conditions considered here. For the pH example, each initial chemical acid unit supplies one hydronium ion. “Strong” describes this conversion, rather than a high concentration. See 14.3 Relative Strengths of Acids and Bases.
An unforgeable reference carrying authority to perform specified operations on an object. In seL4, the operating-system kernel stores security capabilities in protected slots; an integer naming a slot is not itself freely forgeable authority. Guessing another operating-system process's slot number does not give access to its security capability. [ref]
Capabilities tutorial by seL4 Project. Capabilities tutorial, Background: What is a capability?; CNodes and CSlots
An abstract syntax tree can retain source locations even while omitting punctuation nodes. Python records one-based line numbers and zero-based UTF-8 byte offsets, with the end offset exclusive. In
x=2+3*4, the multiplication node spans offsets through , and ast.get_source_segment returns 3*4. A copied location on a folded 12 describes the original span, not text newly present in the source. Such metadata supports error messages and editing tools without making an abstract syntax tree a lossless record of formatting. [ref]ast — Abstract syntax trees, Python 3.14 documentation. Node classes: lineno, col_offset, end_lineno and end_col_offset; get_source_segment and copy_location
// Target: computer-science.bigb
An abstract syntax tree transformation replaces or rearranges nodes. It can simplify a program expression, insert instrumentation, or translate a language construct into simpler constructs. When intended to preserve programming language semantics, it must preserve the relevant observable behavior, including values, side effects, and failures. A changed tree can be structurally valid while changing the program's meaning.
An abstract syntax tree traversal systematically visits its nodes. A depth-first traversal of a tree that processes each child before its parent is called postorder. For the program expression
2+3*4, the schematic postorder is 2, 3, 4, Mult, Add. This order lets an evaluator obtain each operand value before applying its operation. A traversal of the entire Python schema additionally visits marker nodes such as Add and Store; this shorter list concerns only the expression nodes drawn in the figures. Visiting an abstract syntax tree is a structural operation, and is not a general model of execution: a conditional may select only one branch, while a structural visitor examines both.Consider Python programs
x=2+3*4 and x=(2+3)*4. Operator precedence places multiplication inside the right operand of addition in the first abstract syntax tree. The parentheses instead place addition inside the left operand of multiplication in the second. Their right-hand sides evaluate to and . No rule that simply reads all operators left to right would recover both structures. [ref]https://docs.python.org/3.14/reference/expressions.html#operator-precedence. Python language reference, section 6.17, multiplicative operators bind more tightly than additive operators
Two abstract syntax trees for variable assignment: operator precedence gives
. Details. x=2+3*4 the value , while parentheses give x=(2+3)*4 the value 
These are schematic views of Python trees:
Module is omitted, and the Add/Mult and Store markers are written inside their owning nodes. In the actual schema, BinOp has left, op and right fields; Assign has a list of targets and a value. Name(id='x', ctx=Store()) marks the name as a variable assignment target. Reading x in a program expression instead uses Load. A variable assignment is a statement, not an equation asserting that two values are equal. [ref]A restricted recursive evaluator can follow those fields. It accepts only integer literal values and binary addition, subtraction and multiplication; the surrounding variable assignment remains a separate action. Using
type(value) is int excludes Boolean constants, even though Python treats bool as a subclass of int.import ast
OPS = {ast.Add: lambda a, b: a + b,
ast.Sub: lambda a, b: a - b,
ast.Mult: lambda a, b: a * b}
def evaluate(node):
if isinstance(node, ast.Constant) and type(node.value) is int:
return node.value
if isinstance(node, ast.BinOp) and type(node.op) in OPS:
return OPS[type(node.op)](evaluate(node.left), evaluate(node.right))
raise ValueError("outside the integer-expression language")
for source in ("x=2+3*4", "x=(2+3)*4"):
tree = ast.parse(source, optimize=0)
statement = tree.body[0]
print(statement.targets[0].id, evaluate(statement.value))
# x 14
# x 20The children in a rooted tree have ordered roles:
left and right cannot be interchanged. For subtraction, 8-3 gives while 3-8 gives . Operator associativity also matters: 8-3-1 parses as (8-3)-1, giving , whereas 8-(3-1) gives . Here the syntax groups operators to the left; this does not assert the mathematical associativity of subtraction. Operator precedence determines grouping, rather than a general permission to reorder execution. For example, Python evaluates the calls in f()+g()*h() in the order f, g, h, although the multiplication result must be ready before the outer addition. [ref]https://docs.python.org/3.14/reference/expressions.html#evaluation-order. Python language reference, sections 6.16 and 6.17: left-to-right evaluation and left grouping of additive operators
An abstraction function maps a valid concrete representation to its abstract value. Write and . Different storage states can denote the same abstract value. Correct implementation requires both preservation of the representation invariant and agreement with the specified operation after applying the abstraction function; preserving a length bound alone would not prove first-in, first-out behavior.
The representation invariant answers "is this storage state allowed?" The abstraction function answers "which value does it represent?" Thus is a truth value, whereas is a value of the abstract data type. Treating as a partial function on all storage states, or as a function defined on only, expresses the same restriction. The abstraction function may forget storage details, and need not be executable code: its purpose is to connect an implementation to its formal specification.[ref]
Reading 13: Abstraction Functions & Rep Invariants. Rep Invariant and Abstraction Function: representation space, abstract space, partial AF and boolean RI
This distinction makes functional correctness manageable one operation at a time. The unreduced rational representation below separates valid storage from mathematical meaning; correct operations through an abstraction function then checks whether an implementation performs the promised addition. For an example with hidden mutable storage and failure cases, the bounded queue with encapsulation supplies a different abstraction function, reading an occupied prefix of an array as a sequence.
A representation invariant specifies which concrete storage states are valid at the API boundary. Construction must establish it, and each completed operation must preserve it. Internal steps may temporarily violate it only while that state is inaccessible to observers; callbacks or concurrent access would require further reasoning.[ref]
Reading 13: Abstraction Functions & Rep Invariants. Reading 13, How to Establish Invariants: creation, preservation and absence of representation exposure
The short proof pattern is mathematical induction on the number of completed calls: the initial state satisfies ; every permitted transition from an -state ends in an -state; hence every state observable through the API satisfies . An unrestricted external write falls outside this argument.
Abstract Algebra (second edition) by John A. Beachy and others Structure of Groups by 
Codex 0 2026-10-13
Codex 0 2026-10-13Online text: faculty.niu.edu/math_beachy/aaol/structure.shtml
Cited passages: Beachy and Blair, Abstract Algebra, second-edition excerpts, definition 7.2.1 and proposition 7.2.2.
The Beer-Lambert law relates absorbance to an absorbing species' concentration and path length by in a homogeneous medium using sufficiently narrow-band light. The coefficient depends on the species, wavelength and chemical solution conditions. This makes Ellman's reagent useful for quantitative thiol assays once its released colored species has been calibrated. [ref]
Beer–Lambert law. IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry; IUPAC Gold Book: Beer-Lambert law and validity conditions
Here is molar concentration, is the geometric length through the sample, and is the molar absorption coefficient. The ideal beam has parallel light rays, a single wavelength, and a medium with uniform composition and isotropy; finite spectral width must be small relative to features in the optical spectrum.[ref]
Beer–Lambert law. IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry; Beer–Lambert law, entry B00626: opening definition, equations and spectral-bandwidth qualification
For a spectrophotometer, distinguish the source's output from the reference that reaches the photodetector. Let be the received radiant power through a reference blank and the received radiant power through the sample, after subtracting the photodetector's dark response. Both refer to the same selected wavelength band and collection geometry. Define the blank-normalized transmittance and absorbance byThe reference blank includes the solvent and other assay ingredients except the absorbing target. Matching cuvettes and conditions makes their common losses divide out. Thus this is relative to the reference blank; it is not the absolute transmittance of the entire cuvette and its contents. Both radiant powers can be expressed in watts, while their ratio, , and its logarithm, , are dimensionless. An instrument displaying transmission means , not .
Why take a logarithm? Consider two consecutive identical absorbing layers. Each transmits a fraction of the radiant power entering it, so together they transmit . Their absorbances add because . Doubling the sample length doubles absorbance while squaring transmittance. This turns multiplicative loss into an additive scale and makes a straight line useful for inferring molar concentration.
A short model derivation makes the assumption explicit. Suppose a thin layer of thickness removes the fraction of its incoming radiant power, where is a fixed constant of proportionality and is a uniform molar concentration. The corresponding ordinary differential equation, initial condition, and solution areTaking the base-ten logarithm gives , so . The exponential decay follows from equal layers removing equal fractions of what remains. Proportionality to molar concentration additionally assumes that the absorbing chemical species retains the same response as its molar concentration changes; the geometric argument alone does not establish that.
Online text: www.di.ens.fr/~mallat/papiers/WaveletChap7.pdf
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