Sequential model

The sequential model, also known as the Koshland–Némethy–Filmer (KNF) model, is a theoretical framework in biochemistry that describes cooperative interactions in multisubunit proteins, particularly allosteric regulation.^[1] Proposed in 1966 by Daniel E. Koshland Jr., George Némethy, and David Filmer, it posits that the binding of a ligand to one subunit induces a conformational change in that subunit, which sequentially alters the ligand-binding affinity of neighboring subunits through interactions at subunit interfaces.^[1] This induced-fit mechanism allows for a range of intermediate hybrid states, unlike the all-or-nothing transitions in the concerted Monod–Wyman–Changeux (MWC) model, and can account for both positive and negative cooperativity.^[2] The model applies to oligomeric proteins such as hemoglobin and aspartate transcarbamoylase, where ligand binding propagates structural changes across the complex, influencing enzymatic activity or oxygen transport.^[3] It provides a flexible explanation for experimental binding data that the MWC model cannot fully capture, emphasizing the role of local conformational adjustments in allostery.^[1]

Fundamentals

Definition and overview

The sequential model, also known as the Koshland-Némethy-Filmer (KNF) model, is a theoretical framework proposed in 1966 for explaining allosteric cooperativity in multisubunit proteins, where ligand binding to one subunit triggers a localized conformational change that propagates to adjacent subunits.^[4] This model posits that subunits exist in equilibrium between tense (T) and relaxed (R) states, but binding occurs sequentially, with each ligand molecule inducing a shift in the bound subunit's conformation, thereby influencing neighboring subunits' ligand affinity through subunit-subunit interactions.^[4] Unlike models requiring concerted global transitions, the KNF approach emphasizes independent subunit responses that collectively yield cooperative effects, such as sigmoidal binding curves observed in proteins like hemoglobin.^[5] In the sequential model, cooperativity arises from the induced fit mechanism, where ligand binding stabilizes the R state in the affected subunit, enhancing interactions that favor R-state formation in unbound neighbors for positive cooperativity, or conversely stabilizing T states for negative cooperativity.^[4] This framework accounts for allosteric regulation by allowing regulatory molecules to modulate subunit interactions, thereby fine-tuning enzyme activity or ligand binding in response to cellular signals without necessitating symmetry conservation across the entire protein oligomer.^[6] The model's flexibility in describing both homotropic (ligand-specific) and heterotropic (effector-mediated) effects has made it influential in interpreting experimental binding data for oligomeric proteins.^[5] Central to the KNF model is the assumption that subunits are structurally identical and behave independently in the ligand-free state, with conformational changes and cooperative interactions emerging only upon initial ligand engagement, enabling a step-wise assembly of the fully liganded form.^[4] This subunit autonomy until binding distinguishes the model from more rigid symmetry-based alternatives, providing a mechanistic basis for observed asymmetries in allosteric proteins.^[6]

Allosteric cooperativity basics

Allostery refers to the regulation of a protein's activity by the binding of a ligand, known as an effector, at a site distinct from the active site, which induces a conformational change that modulates the protein's function. This mechanism allows for precise control of biological processes, such as enzyme catalysis or receptor signaling, by enabling effectors to either enhance (activation) or inhibit (inhibition) activity without directly competing at the active site.^[7] Cooperativity describes how the binding of one ligand molecule to a protein influences the affinity for additional ligands at other sites, manifesting as positive, negative, or non-cooperative binding. In positive cooperativity, initial ligand binding increases the affinity for subsequent ligands, resulting in a sigmoidal binding curve that reflects heightened sensitivity to ligand concentration changes. Negative cooperativity, conversely, decreases affinity for additional ligands, leading to a binding curve that is less steep than hyperbolic, while non-cooperative binding yields independent sites with a hyperbolic curve. The Hill coefficient (n_H), derived from the Hill equation, quantifies the degree of cooperativity: n_H > 1 indicates positive cooperativity, n_H < 1 negative cooperativity, and n_H = 1 non-cooperativity, with the value approximating the number of interacting sites in highly cooperative systems.^[8] Two primary theoretical frameworks explain allosteric cooperativity: the concerted model, which posits symmetric transitions among subunits, and the sequential model, which allows for stepwise, induced changes in individual subunits. These models provide contrasting views on how conformational dynamics propagate through the protein but both address the observed deviations from simple Michaelis-Menten kinetics in multisubunit systems. Allosteric cooperativity predominantly occurs in multisubunit proteins, such as oligomers (dimers, tetramers, etc.), where the quaternary structure—defined by the spatial arrangement and non-covalent interactions between subunits—facilitates communication between binding sites across the assembly. This oligomeric organization is essential, as it enables ligand-induced perturbations in one subunit to influence distant sites through interfaces involving hydrogen bonds, hydrophobic contacts, and electrostatic interactions.

Development and history

Origins of the model

The study of allostery in the early 1960s was shaped by efforts to explain cooperative ligand binding in multisubunit proteins, building on observations like the Bohr effect in hemoglobin from 1904. By 1965, Jacques Monod, Jeffries Wyman, and Jean-Pierre Changeux introduced the concerted (MWC) model, which posited that all subunits in an oligomeric protein exist in equilibrium between a low-affinity tense (T) state and a high-affinity relaxed (R) state, with ligand binding shifting the entire ensemble concertedly while preserving molecular symmetry.^[9] This symmetry-based framework successfully described positive cooperativity in systems like hemoglobin but faced challenges in accounting for experimental binding data that suggested asymmetric subunit interactions or non-concerted transitions.^[2] These limitations motivated the development of a more flexible alternative that permitted independent conformational changes in individual subunits without requiring global symmetry. The sequential model emerged as a response to the rigidity of the MWC approach, aiming to better explain phenomena such as negative cooperativity—where successive ligand bindings decrease affinity—and binding curves that deviated from predictions of fully symmetric, all-or-none shifts.^[1] By allowing asymmetry and subunit-specific responses, the model addressed discrepancies in empirical data from enzymes and proteins where not all subunits appeared to transition simultaneously.^[2] The sequential model was initially proposed in 1966 by Daniel E. Koshland, Jr., George Némethy, and David Filmer, who compared theoretical predictions to experimental binding isotherms for multisubunit proteins.^[1] This work drew directly from Koshland's earlier induced fit hypothesis, outlined in 1958, which emphasized that enzyme-substrate interactions induce adaptive conformational changes in the protein rather than relying on rigid pre-formed sites. The induced fit concept provided a foundational mechanism for how ligand binding could propagate sequential alterations across subunits, laying the groundwork for the model's emphasis on dynamic, ligand-driven asymmetry in allosteric regulation.^[2]

Key publications and contributors

The sequential model of allosteric regulation was primarily developed by Daniel E. Koshland Jr., George Némethy, and David Filmer in the mid-1960s.^[4] Daniel E. Koshland Jr. (1920–2007), an American biochemist and longtime professor at the University of California, Berkeley, pioneered the induced fit hypothesis in 1958 and extended his work to cooperative binding mechanisms, serving as editor-in-chief of Science from 1985 to 1995.^[10] George Némethy (1934–1994), a Hungarian-born physical chemist who earned his Ph.D. from Cornell University in 1961, specialized in theoretical protein conformation and energetics, contributing computational and energetic analyses to subunit interactions.^[11] David L. Filmer (1932–2021), an American biochemist with a Ph.D. from the University of Wisconsin–Madison in 1961, collaborated on experimental and mathematical aspects of enzyme kinetics and allostery.^[12] The seminal publication introducing the sequential (or induced-fit) model appeared in 1966, titled "Comparison of Experimental Binding Data and Theoretical Models in Proteins Containing Subunits," published in Biochemistry.^[4] In this work, Koshland, Némethy, and Filmer proposed a mechanism where ligand binding to one subunit induces conformational changes that propagate sequentially to adjacent subunits, allowing for both positive and negative cooperativity without requiring global symmetry shifts, as an alternative to the concerted model.^[4] The paper tested the model against experimental binding data, including oxygen-binding isotherms for hemoglobin and ligand-binding curves for enzymes like aspartate transcarbamylase, demonstrating better fits to observed non-sigmoidal and asymmetrical binding patterns than symmetry-constrained theories.^[4] Koshland further refined the sequential model in subsequent years, notably in a 1969 PNAS article titled "Negative Cooperativity in Regulatory Enzymes," where he elaborated how induced-fit mechanisms could explain inhibitory interactions in multi-subunit proteins, supported by binding studies on systems like glyceraldehyde-3-phosphate dehydrogenase.^[13] His later reviews, such as those tracing the evolution of induced fit from template theories to dynamic conformational models, underscored the model's enduring relevance in interpreting allosteric data from the 1960s onward.

Core principles

Induced fit hypothesis

The induced fit hypothesis was first proposed by Daniel E. Koshland Jr. in 1958 as an alternative to the rigid template model of enzyme-substrate interactions. In this model, the enzyme is not a static structure with a preformed active site that perfectly matches the substrate, but rather a flexible entity whose active site undergoes a conformational change upon substrate binding to achieve optimal alignment of catalytic groups for the reaction. This induced adjustment ensures specificity and efficiency, explaining why certain substrate analogs bind but fail to catalyze if they do not trigger the necessary structural rearrangement. This hypothesis directly contrasts with the lock-and-key model introduced by Emil Fischer in 1894, which posited a rigid, complementary fit between enzyme and substrate akin to a key entering a lock. Koshland's induced fit emphasizes the dynamic nature of proteins, supported by evidence such as reversible denaturation experiments showing structural flexibility in enzymes exposed to urea, where viscosity and optical rotation changes indicate malleable conformations. By incorporating flexibility, the induced fit model better accounts for phenomena like differential reaction rates among similar substrates, where poor inducers fail to align catalytic residues despite surface complementarity. In the context of the sequential model for allosteric proteins, developed by Koshland, George Némethy, and David L. Filmer in 1966, the induced fit mechanism underpins cooperative binding in multisubunit ensembles. Here, ligand binding to one subunit triggers a local conformational shift from a low-affinity tense (T) state to a high-affinity relaxed (R) state, which sequentially influences neighboring subunits by altering their binding sites through subunit interactions. This propagation allows for graded responses in affinity across the oligomer, extending the original induced fit concept from single-site enzymes to allosteric regulation in proteins like those exhibiting homotropic or heterotropic effects. The change in binding affinity is quantified by association constants K_T for the T state and K_R for the R state, where K_R > K_T reflects the enhanced affinity post-induction in cases of positive cooperativity.

Rules of the KNF model

The Koshland-Némethy-Filmer (KNF) sequential model is defined by two core assumptions that govern ligand binding in multisubunit proteins. First, the binding of a ligand to one subunit induces a conformational change specifically in that subunit. Second, this induced change alters the interactions between the bound subunit and its adjacent subunits, thereby modifying the affinity of neighboring subunits for subsequent ligand molecules.^[4] These assumptions are formalized through the i³ criteria, which specify the conditions necessary for cooperative interactions in the model: (i) the subunits are identical, (ii) ligand binding induces conformational changes within individual subunits, and (iii) intramolecular interactions between subunits are affected by these changes, influencing binding at other sites. This framework ensures that cooperativity arises from sequential, subunit-specific transitions rather than global shifts. Mathematically, the KNF model adapts the Adair equation to describe sequential binding steps, where the fractional saturation Y of a protein with n binding sites is given by

Y = \frac{\sum_{i=1}^{n} i \beta_i [L]^i}{\sum_{i=0}^{n} \beta_i [L]^i},

with \beta_i = \prod_{j=1}^{i} K_j representing the product of stepwise association constants K_j for the j-th ligand binding, and [L] the ligand concentration; the factor i accounts for the number of occupied sites in the i-ligand complex. These constants K_1, K_2, \dots, K_n can vary to reflect changes in affinity due to prior bindings.^[4] The rules of the KNF model permit asymmetry in subunit conformations, as each subunit can independently adopt a changed state upon ligand binding without requiring synchronous transitions across the protein. This flexibility allows for variable cooperativity, where the degree of positive or negative cooperativity depends on the specific interaction parameters between subunits, enabling the model to fit a wide range of experimental binding curves.^[4]

Mechanisms and features

Sequential conformational changes

In the sequential model of allosteric cooperativity, ligand binding to a multisubunit protein induces a conformational change in the affected subunit from the tense (T) state to the relaxed (R) state, which in turn modifies the ligand-binding affinity of adjacent subunits through direct structural interactions. This stepwise mechanism allows for progressive propagation of the conformational shift across the protein oligomer, where each binding event locally alters the subunit's environment, facilitating subsequent bindings without requiring a simultaneous global transition. These changes ensure that the conformational adjustment in one subunit influences neighboring ones.^[4] Unlike models that enforce overall symmetry, the sequential approach permits unbound subunits to retain their original T conformation even as bound subunits adopt the R state, leading to asymmetric hybrid intermediates during the binding process. For a tetrameric protein, this results in a series of distinct states, such as one R and three T subunits (T₃R), two R and two T (T₂R₂), and so on, each with varying stability based on intersubunit contacts. These hybrid states are stabilized by the cooperative effects of the induced changes, allowing the protein to exist in partially liganded forms that bridge the fully unliganded T₄ and fully liganded R₄ conformations.^[4] The binding scheme for a tetramer in this model involves sequential steps with intrinsically increasing association constants, reflecting enhanced affinity as more subunits transition to the R state. For instance, the first ligand binds with association constant K₁ to a T subunit, yielding the T₃R hybrid; the second binds preferentially to an adjacent T subunit in this hybrid with K₂ > K₁, forming T₂R₂; subsequent bindings follow with K₃ > K₂ and K₄ > K₃, culminating in the R₄ state. This progression arises from the conformational changes that strengthen binding sites in neighboring subunits, enabling positive cooperativity through localized rather than wholesale structural rearrangements.^[4]

Support for negative cooperativity

In the Koshland-Némethy-Filmer (KNF) sequential model, negative cooperativity arises when ligand binding to one subunit induces a conformational change that propagates inhibitory effects to adjacent subunits through repulsive or energetically unfavorable subunit-subunit interactions, thereby reducing the binding affinity at unoccupied sites. This mechanism allows for asymmetric changes within the oligomer, where the altered conformation of a bound subunit hinders ligand association elsewhere, contrasting with facilitative interactions that promote positive cooperativity.^[4] This affinity reduction is quantitatively described by successive association constants that decrease with each binding event, such as K_3 < K_2 in a trimeric protein, where K_n represents the equilibrium constant for the n-th ligand binding step. Such diminishing affinities result in binding curves that deviate negatively from hyperbolic, producing concave Hill plots with a Hill coefficient n_H < 1, indicating diminished responsiveness to ligand concentration as saturation progresses.^[4] A key advantage of the KNF model over the Monod-Wyman-Changeux (MWC) concerted model is its inherent support for negative cooperativity; the MWC framework, which requires all subunits to transition simultaneously between tense and relaxed states, cannot generate this phenomenon without ad hoc modifications, whereas the sequential induced-fit process in KNF accommodates it directly through subunit-specific interactions.^[14] Pre-2017 experimental studies on multisubunit enzymes provided support for this inhibitory propagation, particularly through observations of half-the-sites reactivity, where ligand or substrate occupancy is limited to approximately half the available sites even under saturating conditions, reflecting the model's prediction of strong negative interactions that deactivate remaining sites. For instance, kinetic analyses of cytidine triphosphate synthetase demonstrated this reactivity, aligning with sequential conformational inhibition rather than symmetric binding.^[5]

Applications in proteins

Hemoglobin as a case study

Hemoglobin (HbA) is a tetrameric protein composed of two α subunits and two β subunits arranged in an α₂β₂ configuration, with each subunit containing a heme group capable of binding one oxygen molecule.^[15] The protein exists in two primary conformational states: the low-affinity tense (T) state in its deoxy form and the high-affinity relaxed (R) state upon oxygenation.^[16] In the Koshland-Némethy-Filmer (KNF) sequential model, oxygen binding to hemoglobin proceeds stepwise, with the initial O₂ molecule binding preferentially to one of the α subunits in the T state, inducing a local conformational change from a low-affinity (A) to a high-affinity (B) state in that subunit.^[17] This induced fit alters subunit interfaces, facilitating subsequent binding to an adjacent β subunit by increasing its affinity through interactions such as changes in salt bridges and heme geometry.^[4] Successive bindings propagate these changes across the tetramer, leading to a hybrid structure with mixed A and B conformations until the fully oxygenated R state is reached.^[4] The cooperative nature of this process is reflected in the intrinsic stepwise association constants, which increase progressively: K_1 \approx 0.01 mmHg^{-1} for the first binding step and K_4 \approx 5 mmHg^{-1} for the fourth, illustrating how early low-affinity bindings give way to high-affinity ones in later steps.^[18] Although the sequential model accounts for the asymmetric, induced conformational transitions observed in hemoglobin, it requires a complex set of interaction parameters to fit binding data, and the Monod-Wyman-Changeux (MWC) concerted model is often preferred due to hemoglobin's demonstrated symmetry in quaternary structure changes between T and R states.^[19]

Other enzymes and proteins

The sequential model has been applied to phosphofructokinase (PFK), a key regulatory enzyme in glycolysis that exhibits cooperative binding of fructose-6-phosphate (F6P). In bacterial PFK from Bacillus stearothermophilus, the binding of F6P to one subunit induces a conformational change that propagates asymmetrically to adjacent subunits, increasing their affinity for F6P and resulting in positive cooperativity, consistent with the induced-fit mechanism of the KNF model. This sequential process explains the enzyme's sigmoidal kinetics, where ATP acts as an allosteric inhibitor by binding to a distinct site and stabilizing a low-affinity conformation in unliganded subunits. Tyrosyl-tRNA synthetase, a dimeric enzyme involved in protein synthesis, exemplifies negative cooperativity under the sequential model. Binding of one tRNA^Tyr molecule to a subunit induces a conformational shift that reduces the affinity of the second subunit for another tRNA^Tyr, leading to half-of-the-sites reactivity where only one site is typically occupied at saturation.^[20] This asymmetric response, which the concerted MWC model cannot accommodate, aligns with the KNF framework's allowance for independent subunit transitions that diminish subsequent ligand binding.^[20] G-protein coupled receptors (GPCRs), despite being largely monomeric, display allosteric behaviors resembling the sequential model through domain-like structural elements and induced-fit propagation. Ligand binding at the orthosteric site triggers sequential conformational changes across the seven-transmembrane helices, such as outward movement of transmembrane helix 6, which propagates allosterically to modulate G-protein coupling or downstream signaling.^[21] This stepwise induced fit, involving both conformational selection and structural adjustments, facilitates signal transduction in GPCRs like the β₂-adrenergic receptor, where asymmetry in domain rearrangements supports graded activation.^[22] Post-2017 computational studies, including molecular dynamics simulations, have validated the sequential model's emphasis on asymmetric allostery in diverse proteins by revealing ligand-induced propagations of structural changes that favor hybrid states over symmetric transitions.^[23]

Comparisons with alternative models

The MWC concerted model

The Monod–Wyman–Changeux (MWC) model, proposed in 1965 by Jacques Monod, Jeffries Wyman, and Jean-Pierre Changeux, posits that allosteric proteins composed of identical subunits exist in an equilibrium between two conformational states: a low-affinity tense (T) state and a high-affinity relaxed (R) state. Upon ligand binding, the entire oligomer undergoes a concerted transition from the T to the R state, with all subunits changing conformation simultaneously to maintain molecular symmetry. This mechanism accounts for cooperative binding without requiring subunit-specific interactions.^[24] Key features of the MWC model include the preservation of quaternary symmetry throughout the transition, as the protein oligomer switches entirely between the symmetric T and R forms. Ligands bind with higher affinity to the R state, thereby stabilizing it and shifting the T/R equilibrium toward the R form, which enhances binding at remaining sites and promotes positive cooperativity. Importantly, the model excludes hybrid intermediates where only some subunits are in the R state, emphasizing a symmetric, all-or-nothing conformational switch.^[24] The mathematical foundation of the MWC model derives from statistical mechanics applied to the binding equilibria of the T and R states. For a protein with n identical subunits and ligand concentration [S], the fractional saturation Y is given by:

Y = \frac{\alpha (1 + \alpha)^{n-1} + L c \alpha (1 + c \alpha)^{n-1}}{(1 + \alpha)^n + L (1 + c \alpha)^n}

where \alpha = [S]/K_R (with K_R as the dissociation constant for the R state), L = [T_0]/[R_0] (the allosteric constant representing the T/R equilibrium in the absence of ligand), and c = K_R / K_T (the ratio of dissociation constants, typically c < 1 since K_T > K_R). This equation captures how the parameters L and c govern the degree of cooperativity.^[24] The MWC model's strength lies in its ability to quantitatively explain positive cooperativity in symmetric multisubunit systems, such as hemoglobin, where oxygen binding induces a concerted shift that facilitates subsequent bindings and produces a sigmoidal saturation curve consistent with experimental oxygen-binding data.

Key differences: structural and functional

The sequential (KNF) model and the concerted (MWC) model differ fundamentally in their structural predictions for allosteric proteins. In the KNF model, ligand binding induces tertiary conformational changes in individual subunits, allowing for asymmetric hybrid states where not all subunits adopt the same conformation simultaneously.^[6] This induced-fit mechanism permits subunit-specific adjustments without requiring global symmetry. In contrast, the MWC model mandates strict symmetry across all subunits in any given state, with allosteric transitions involving large quaternary structural rearrangements that shift the entire oligomer between tense (T) and relaxed (R) conformations in an all-or-none fashion.^[2] These structural distinctions mean the KNF model accommodates mixed-affinity states, while the MWC model precludes them, enforcing uniform subunit behavior.^[25] Functionally, the KNF model's allowance for asymmetry enables it to explain negative cooperativity and half-of-the-sites reactivity, where binding at one site reduces affinity at others due to induced steric or energetic interactions between subunits.^[26] For instance, it accounts for scenarios where only half the sites in a dimer or tetramer are occupied at saturation, a phenomenon incompatible with the MWC model's symmetric, equivalent subunits. The MWC model, however, is inherently limited to positive cooperativity or non-cooperative binding, as its concerted transitions favor enhanced affinity across all sites upon the T-to-R shift, without mechanisms for inhibitory subunit interactions.^[2] Thus, the KNF model provides greater explanatory power for diverse regulatory behaviors, including inhibition, while the MWC model excels in describing amplification of binding signals in systems like oxygen transport.^[5] Experimental distinctions between the models arise in binding curve analyses and structural data. The KNF model fits asymmetric binding data, such as gradual saturation curves reflecting sequential subunit activation, as seen in enzymes exhibiting negative homotropic effects.^[25] For example, the aspartate chemoreceptor shows a 20-fold affinity difference between binding sites in half-liganded states, with crystallographic evidence of asymmetric tertiary shifts supporting sequential changes.^[6] In contrast, hemoglobin's sigmoidal oxygen-binding curve and symmetric intermediates favor the MWC model, where quaternary rotations (e.g., 15° α1β2 interface shift) occur without detectable asymmetric hybrids. Enzymes like glutamine synthetase, displaying negative cooperativity in metal ion binding, align better with KNF predictions due to their non-sigmoidal curves and site-specific affinities.^[27] Ongoing debates highlight that neither model fully captures allosteric complexity, with computational studies blending elements of both. For instance, molecular dynamics simulations of hemoglobin reveal networks of tertiary and quaternary motions that challenge pure MWC symmetry while incorporating KNF-like induced fits, suggesting hybrid mechanisms without resolving the dichotomy.