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, and by additional, direct protein-protein interactions . The domain formation is closely linked to T-cell activation, with TRC clusters forming within seconds of T-cell adhesion triggering the first activation signals |
Direct evidence for a central role of the length of receptor-ligand complexes comes from experiments in which these lengths are altered by protein engineering |
. Milstein and coworkers have considered variants of the protein CD48 with four and five immunoglobolin-like (Ig-like) domains. The CD48 variants are longer than the CD48 wildtype, which contains only two Ig-like domains. The CD48 wildtype and both CD48 variants bind to CD2 on T cells. From electron micrographs of the ... |
In this article, we calculate detailed phase diagrams for cells adhering to supported membranes with anchored ligands. We consider two general adhesion scenarios: In the first scenario, long and short ligands in the supported membrane bind to the same cell receptor (see section 4), as in the experiments of Milstein and... |
, in which CD48 wildtype and a CD28 variant in the supported membrane both bind to CD2 in the T-cell membrane. In the second scenario, two types of ligands in the supported membrane bind to two types of receptors in the cell membrane (see section 5), as in experiments in which MHCp and ICAM1 in the supported membrane b... |
Our calculations are based on a statistical-physical model of cell adhesion (see section 2). In this model, the membranes are described as elastic sheets discretized into small patches that can contain single receptor or ligand molecules |
The binding and domain formation of receptor-ligand complexes is affected by thermally excited shape fluctuations of the membranes on nanometer scales. These shape fluctuations lead to a critical point for the segregation of long and short receptor-ligand complexes |
. The critical point depends on the length difference of the complexes, on the concentrations and affinities of the receptors and ligands, and on the bending rigidity of the membranes (see section 4). The critical point constitutes a threshold for segregation, or domain formation, and may help to understand why Milstei... |
. In addition, the membrane shape fluctuations on nanoscales lead to a cooperative binding of receptor-ligand complexes (see section 3). |
Statistical-physical description of cell adhesion Cell adhesion involves length scales that differ by orders of magnitude (see fig. ). The diameters of the cell and cell contact zone have values of several micrometers, while the average separation of the membranes within the contact zone is typically tens of nanometers... |
The binding equilibrium and segregation of receptor-ligand complexes in cell contact zones is affected by membrane shape deformations and fluctuations. Since bound receptor-ligand complexes constrain the local separation of the membranes, the relevant deformations and fluctuations of the membranes occur on lateral leng... |
. It is reasonable to assume that the elasticity of the membranes is dominated by their bending rigidity on these length scales. The binding rigidity [MATH] dominates over the membrane tension [MATH] on lateral length scales smaller than the crossover length [MATH] |
, which is of the order of several hundred nanometers for cell membranes . The cytoskeletal elasticity contributes on length scales larger than the average distance between the cytoskeletal anchors in the membrane, which may be around 100 nanometers |
. The bending rigidity thus is likely to dominate over the lateral tension and the cytoskeletal elasticity on lateral length scales up to 100 nanometers relevant here. |
We have developed discrete models for the adhesion of membranes via anchored receptors and ligands . In discrete models, the two apposing membranes in the contact zone of cells or vesicles are divided into small patches |
. In our models, the rigidity-dominated elasticity of the membranes in the contact zones of cells or vesicles is described by [EQUATION] |
where [MATH] is the local separation of the apposing membrane patches [MATH] . The elastic energy depends on the mean curvature [MATH] of the separation field [MATH] with the discretized Laplacian [MATH] . Here [MATH] to [MATH] are the membrane separations at the four nearest-neighbor patches of membrane patch [MATH] o... |
The ‘effective bending rigidity’ of the two membranes with rigidities [MATH] and [MATH] is [MATH] . If one of the membranes, e.g. membrane 2, is a planar supported membrane, the effective bending rigidity [MATH] equals the rigidity [MATH] of the apposing membrane since the rigidity [MATH] of the supported membrane is t... |
The overall energy of the membranes in the cell contact zone [EQUATION] is the sum of the elastic energy [MATH] and interaction energy [MATH] . The interaction energy depends on the distribution [MATH] of the receptors in membrane 1, on the distribution [MATH] of the receptors in membrane 2, and on the separation field... |
Adhesion via a single type of receptor-ligand complexes 3.1 Interaction energy of receptors and ligands We first consider the case in which the adhesion is mediated by a single type of receptor-ligand complexes. Examples of this case are (i) cells adhering to supported membranes that contain a single type of ligand |
, and (ii) vesicles with anchored receptors that adhere to supported membranes or surfaces with complementary ligands The interactions of receptors and ligands within the contact zone of the cell or vesicle are described by the interaction energy |
[EQUATION] in our model. Here, the occupation number [MATH] or 0 indicates whether a receptor is present or absent in membrane patch [MATH] of the cell, and [MATH] or 0 indicates whether a ligand is present or absent in patch [MATH] of the apposing membrane. Receptor and ligand molecules in apposing patches [MATH] of t... |
[EQUATION] which depends on the binding energy [MATH] , and the length [MATH] and binding width [MATH] of a receptor-ligand complex. A receptor thus binds to an apposing ligand with energy [MATH] if the local separation [MATH] of the membranes is within the binding range [MATH] |
3.2 Effective adhesion potential The binding equilibrium of the membranes in the contact zone can be determined from the free energy [MATH] , where [MATH] is the partition function of the system, [MATH] is Boltzmann’s constant, and [MATH] is the temperature. The partition function [MATH] is the sum over all possible me... |
[EQUATION] For typical concentrations of receptors and ligands in cell adhesion zones up to hundred or several hundred molecules per square micrometer, the average distance between neighboring pairs of receptor and ligand molecules is much smaller than the width of the molecules. For these small concentrations, the eff... |
[EQUATION] where [MATH] is the area concentration of unbound receptors in the cell membrane, and [MATH] is the area concentration of unbound ligands in the apposing membrane. The binding equilibrium in the contact zone thus can be determined from considering two membranes with the elastic energy ( ) that interact via a... |
3.3 Area fraction [MATH] of the membranes within binding range of receptors and ligands Receptor-ligand complexes can only form at membrane patches with a local separation within the binding range [MATH] of the receptors and ligands (see eq. ( )). The area concentration [MATH] of the receptor-ligand complexes in the co... |
[EQUATION] Here, [MATH] is the equilibrium constant for receptor-ligand binding within this membrane fraction. In our model, the equilibrium constant is [MATH] |
In equilibrium, the fraction [MATH] of membrane patches with a local separation within receptor-ligand binding range depends on the effective binding energy [MATH] , the binding width [MATH] , the effective rigidity [MATH] of the membranes, and the temperature [MATH] . We have found that the effect of these four quanti... |
[EQUATION] To a first approximation, the membrane fraction [MATH] depends only on [MATH] for typical lengths and concentrations of receptor-ligand complexes in cell adhesion zones. In cell adhesion zones, direct contacts between the membranes can be neglected since the average separation of the membranes, which depends... |
From Monte Carlo simulations, we have found that the functional dependence of the area fraction [MATH] on the rescaled potential depth [MATH] is well described by |
[EQUATION] with the dimensionless coefficient [MATH] . The membrane fraction [MATH] increases with [MATH] and, thus, increases with the effective binding energy [MATH] and the effective bending rigidity [MATH] . The reason for this increase is that the roughness of the membranes resulting from thermal shape fluctuation... |
3.4 Concentrations of bound and unbound receptors of an adhering cell From eqs. ( ) to ( ), we obtain the relation [EQUATION] between the area concentration [MATH] of bound receptor-ligand complexes in the contact zone and the area concentrations [MATH] and [MATH] of unbound receptors and ligands. This nonlinear relati... |
The total number [MATH] of receptors in the cell membrane is constant. The concentrations of bound and unbound receptors are therefore connected by the additional relation |
[EQUATION] where [MATH] is the total area of the cell membrane, and [MATH] the contact area. We have neglected here the area occupied by bound receptor-ligand complexes since this area is small compared to the total contact area [MATH] for typical concentrations in cell adhesion zones. The concentrations of unbound rec... |
Two types of membrane-anchored ligands adhering to the same cell receptor 4.1 Interaction energy of receptors and ligands In recent experiments by Milstein and coworkers |
, long and short ligands anchored to a supported membrane bind to the same receptor of an adhering T cell. These ligands are wildtype CD48 and elongated CD48 variants, and the receptor in the T cell membrane is CD2. In our model, this situation is described by the interaction energy |
[EQUATION] for the two apposing membranes in the cell contact zone. Here, the occupation number [MATH] , 2, or 0 indicates whether a ligand [MATH] of type 1, a ligand [MATH] of type 2, or no ligand is present in patch [MATH] of the supported membrane, and [MATH] or 0 indicates whether a receptor [MATH] is present or no... |
[EQUATION] and [EQUATION] with binding energies [MATH] and [MATH] and equilibrium lengths [MATH] of the complexes [MATH] and [MATH] . We have assumed here that the two complexes have the same binding width [MATH] |
4.2 Effective adhesion potential As in section 3.2 , the summations over all possible distributions [MATH] and [MATH] of receptors and ligands in the partition function of the model leads to an effective adhesion potential |
. The effective adhesion potential now is a double-well potential (see fig. ). Both wells have the same width [MATH] as the potentials ( 12 ) and ( 13 ). The well with its center at the membrane separation [MATH] reflects the interactions of the receptors [MATH] with the shorter ligands [MATH] , and the well centered a... |
[EQUATION] and [EQUATION] depend on the concentrations [MATH] [MATH] and [MATH] of unbound receptors and ligands, and on the binding constants [MATH] and [MATH] for receptors and ligands within the appropriate binding ranges |
. The binding equilibrium of the receptors [MATH] and ligands [MATH] and [MATH] in the contact zone thus can be determined from considering two apposing membranes with elastic energy ( ) that interact via an effective double-well potential with well depths [MATH] and [MATH] given by eqs. ( 14 ) and ( 15 ). |
4.3 Phase diagram If the two wells of the effective adhesion potential are relatively shallow, thermal membrane fluctuations can easily drive membrane segments to cross from one well to the other. If the two wells are deep, the crossing of membrane segments from one well to the other well is hindered by the potential b... |
. Beyond a critical depth of the potential wells, the line tension leads to the formation of large membrane domains that are bound in well one or well two. Within each domain, the adhesion of the membranes is predominantly mediated either by the receptor-ligand complexes [MATH] or by the complexes [MATH] |
We have previously found that the critical potential depth for domain formation is [EQUATION] with the prefactor [MATH] determined by Monte Carlo simulations |
. Domain formation in the contact zone or, in other words, segregation of the complexes [MATH] and [MATH] can only occur if the effective potential depths [MATH] and [MATH] exceed the critical potential depth [MATH] . The critical potential depth depends on the temperature [MATH] and the bending rigidity [MATH] as well... |
. For these complex concentrations and lengths, the thermal membrane roughness is smaller than the lengths of the receptor-ligand complexes. |
Domain coexistence occurs for equal depths [EQUATION] of the potential wells if the two wells have the same width [MATH] as in Fig. . With eqs. ( 14 ) and ( 15 ), this coexistence condition implies that domain coexistence occurs along the line with |
[EQUATION] in the [MATH] [MATH] -plane. The line has the slope [MATH] and ends at a critical point (see phase diagram in fig. (a)). For [MATH] , we have [MATH] . The adhesion is then dominated by the short complexes [MATH] throughout the cell contact zone. For [MATH] , in contrast, the adhesion is dominated by the long... |
, see Discussion. Two types of membrane-anchored ligands adhering to different cell receptors 5.1 Interaction energy of receptors and ligands and effective adhesion potential |
Several experimental groups have investigated the adhesion of T cells to supported membranes with anchored MHCp and ICAM1 ligands |
. The ligand MHCp binds to the T cell receptor (TCR), and the ligand ICAM1 to the integrin LFA1 in the T cell membrane. The TCR-MHCp complex has a length of around 13 nm |
, and the LFA1-ICAM1 complex a length of 40 nm . A situation in which two ligands in the supported membrane bind to different receptors in a cell membrane can be described in our model via the interaction energy |
[EQUATION] Here, the occupation number [MATH] , 2, or 0 indicates whether a receptor [MATH] , a receptor [MATH] , or no receptor is present in patch [MATH] of the cell membrane in the contact zone, while [MATH] , 2, or 0 indicates whether a ligand [MATH] , a ligand [MATH] , or no ligand is present in the apposing patch... |
[EQUATION] and [EQUATION] now depend on the concentrations [MATH] and [MATH] of unbound receptors in the cell membrane, on the concentrations [MATH] and [MATH] of unbound ligands in the supported membrane, and on the binding constants [MATH] and [MATH] for the complexes [MATH] and [MATH] |
5.2 Phase diagram As in section 4.3 , domain coexistence in the cell contact zone requires equal depths [MATH] of the potential wells if the two wells have the same width [MATH] . The effective adhesion potential then is a symmetric double-well potential. We assume here again that the total area of the supported membra... |
For the symmetric double-well potential with [MATH] , the membrane fractions [MATH] and [MATH] within well 1 and well 2 depend primarily on the rescaled potential depth |
[EQUATION] as in section 3.3 . The Monte Carlo data in fig. (a) illustrate how [MATH] and [MATH] depend on [MATH] . Below the critical potential depth [MATH] , the membrane fluctuates between the two wells. Because of the symmetry of the potential, [MATH] and [MATH] attain the same value [MATH] for [MATH] . Above the c... |
Since the total numbers [MATH] and [MATH] of receptors 1 and 2 in the cell membrane are constant, we have [EQUATION] Here, [MATH] is the total area of the cell, [MATH] the contact area, and [MATH] is the fraction of the contact area occupied by domain 1, which is predominantly bound in well 1. The first terms on the ri... |
With the four independent, dimensionless parameters [EQUATION] and [EQUATION] the eqs. ( 22 ) and ( 23 ) can be rewritten as [EQUATION] |
since we have [MATH] (see eqs. ( 19 ) to ( 21 )). From these two equations, one can determine [MATH] and [MATH] as functions of the independent parameters [MATH] [MATH] [MATH] and [MATH] . Domain coexistence in the cell contact zone occurs for [MATH] and [MATH] |
To obtain general relations for the critical point and the boundary lines of the two-phase region in the [MATH] [MATH] plane, we first solve eq. ( 26 ) for [MATH] and eq. ( 27 ) for [MATH] , which leads to |
[EQUATION] At the critical point, we have [MATH] . By inserting these relations into eqs. ( 28 ) and ( 29 ), we obtain a general expression for the location [MATH] of the critical point in the [MATH] [MATH] plane: |
[EQUATION] For the Monte Carlo data of fig. (a) with [MATH] and [MATH] , for example, we have [MATH] . Since [MATH] and [MATH] have to be positive, domain coexistence can only occur if [MATH] and [MATH] are both larger than [MATH] . The domain-coexistence region in the [MATH] [MATH] plane is bounded by two lines with [... |
[EQUATION] for the [MATH] line. Similarly, inserting [MATH] in the eqs. ( 28 ) and ( 29 ) leads to the parametric form [EQUATION] |
for the [MATH] line. The domain-coexistence region in the phase diagram of fig. (b), for example, follows from inserting the functions |
[MATH] and [MATH] obtained from interpolation of the Monte Carlo data shown in fig. (a) into eqs. ( 31 ) and ( 32 ). For [MATH] , we have [MATH] . The [MATH] line then is given by |
[EQUATION] since we have [MATH] for [MATH] . Similarly, the [MATH] line is given by [EQUATION] because of [MATH] . For [MATH] , the membranes are only bound via one of the wells. The membrane fraction [MATH] bound in this well therefore can be approximated by the same expression [MATH] with [MATH] as in the case of an ... |
The [MATH] line has a vertical asymptote in the [MATH] [MATH] plane, since [MATH] in eq. ( 33 ) diverges for [EQUATION] because the denominator of the right-hand side of eq. ( 29 ) vanishes. With [MATH] , we obtain the location |
[EQUATION] for this vertical asymptote from eq. ( 35 ). Similarly, the [MATH] line has a horizontal asymptote in the [MATH] [MATH] plane since the the denominator of the right-hand side of eq. ( 34 ) vanishes for |
[EQUATION] With [MATH] , we obtain the value [EQUATION] for the horizontal asymptote of the [MATH] line from eq. ( 37 ). Discussion and Conclusions |
In this article, we have determined phase diagrams for cells adhering to supported membranes with anchored ligands. For supported membranes with short and long ligands [MATH] and [MATH] that bind to the same cell receptor [MATH] , coexistence of domains of [MATH] and [MATH] complexes in the cell adhesion zone only occu... |
). A coexistence line as in the diagram of fig. (a)) is typical for phase diagrams in grand-canonical ensembles with constant chemical potentials. |
For supported membranes with two types of ligands [MATH] and [MATH] that bind to different cell receptors [MATH] and [MATH] , we obtain a qualitatively different phase diagram with a broad coexistence region (see fig. (b)). Domain coexistence in the cell adhesion zone occurs for [MATH] . The broad coexistence region is... |
Milstein and coworkers have observed domain coexistence for a narrow concentration ratio of short and long ligands that bind to the same cell receptor CD2, in agreement with our phase diagram in fig. (a). However, two differences between our phase diagram in fig. (a) and the phase diagram of Milstein and coworkers in f... |
are: First, the coexistence line in the phase diagram of Milstein and coworkers seems to have a finite width. Such a finite width may result from slight changes of the ligand concentrations upon binding, since several cells adhere to the same supported membrane in the experiments. Second, the coexistence line in the di... |
. We obtain a phase diagram similar to the diagram of Milstein and coworkers if the well depths [MATH] and [MATH] at which the membranes unbind are larger than the critical potential depth [MATH] , which determines the location of the critical point in the diagram of fig. (a). |
We find that thermal membrane shape fluctuations on nanometer scales play a central role during cell adhesion. Fluctuations on these scales have been recently reported for immune cells adhering to coated substrates |
. In previous work, we have found that the fluctuations lead to a cooperative binding of receptors and ligands (see fig. , and to a critical point for the segregration of long and short receptor-ligand complexes |
. Our phase diagrams in fig. are therefore qualitatively different from phase diagrams calculated under neglection of shape fluctuations |
. The binding cooperativity of receptors and ligands arises since a receptor-ligand complex locally constrains the membrane shape fluctuations and facilitates the binding of nearby complexes. The binding cooperativity is thus closely related to the fluctuation-induced attractive interactions between bound receptor-liga... |
, which result from a suppression of membrane-shape fluctuations, similar to the fluctuation-induced interactions of rigid membrane inclusions |
We have neglected here the line tension of the domain boundaries, which may suppress the formation of small domains in the cell adhesion zone. In classical nucleation theory, the line tension leads to a threshold size for stable domains. Experimental observations of stable microdomains in the adhesion zones of immune c... |
indicate that this threshold size is rather small. We will consider the line tension between domains of short and long receptor-ligand complexes in detail in a future article. Bibliography |
# Source: arxiv 1008.0298 # Title: Distribution of dwell times of a ribosome: effects of infidelity, kinetic proofreading and ribosome crowding # Sections: all # Downloaded: 2026-03-03T05:15:23.827586+00:00 |
Distribution of dwell times of a ribosome: effects of infidelity, kinetic proofreading and ribosome crowding Abstract Ribosome is a molecular machine that polymerizes a protein where the sequence of the amino acid residues, the monomers of the protein, is dictated by the sequence of codons (triplets of nucleotides) on ... |
pacs: 87.16.ad, 87.16.Nn, 87.10.Mn Introduction The primary structure of a protein consists of a sequence of amino acid residues linked together by peptide bonds. Therefore, a protein is also referred to as a polypeptide which is essentially a linear hetero-polymer, the amino acid residues being the corresponding monom... |
spirinbook spirin02 spirin09 and the process is referred to as translation (of genetic code). The polymerization of protein takes places in three stages that are identified as initiation elongation (of the protein) and |
termination The ribosome also utilizes the template mRNA as a track for its own movement; it steps forward by one codon at a time while, simultaneously, it elongates the growing polypeptide by one amino acid monomer. Therefore, ribosome is also regarded as a motor that, like other molecular motors, takes input (free-) ... |
Enormous progress has been made in the last decade in the fundamental understanding of the structure, energetics and kinetics of ribosomes |
ramanobel adanobel steitznobel steitznature ramanature ramacell frank06a frank06b frank09 frank10 Recent single molecule studies of ribosomes |
marshall08 munro08 munro09 blanchard09 blanchard04 uemura10 aitken10 has thrown light on its operational mechanism. In single molecule experiments, it has been observed that a ribosome steps forward in a stochastic manner; its stepping is characterized by an alternating sequence of pause and translocation. The sum of t... |
The probability density [MATH] of the dwell times of a ribosome, measured in single-molecule experiments wen08 , has been compared with the corresponding data obatined from computer simulations wen09 . It has been claimed that the data do not fit a single exponential thereby indicating the existence of more than one ra... |
A systematic analytical derivation of the DTD of the ribosomes was presented recently gccr from a kinetic theory of translation basu07 that also involves essentially five steps in the mechano-chemical cycle of a ribosome. However, the model developed in ref. basu07 ignores some of the key features of the mechano-chemic... |
basu07 by capturing the processes of proofreading and allowing for the possibility of imperfect fidelity of translation. Moreover, the identification of the mechano-chemical states as well as the nature of the transitions among these states is revised in the light of the empirical facts established in the last couple o... |
sharmathesis , we analytically calculate the probability density [MATH] of the dwell times of ribosomes. The DTD derived in this paper is qualitatively similar to that observed by Wen et al. in their single molecule experiments wen08 However, because of the sequence inhomogeneity of the template mRNA used in the experi... |
Interestingly, the inverse of the mean dwell time is the average velocity [MATH] of the ribosome motor. Since, in our model, the ribosome is not allowed to back track on the mRNA template, [MATH] is also the mean rate of elongation of the polypeptide. We show that [MATH] satisfies an equation that resembles the Michael... |
It is well known that most often a large number of ribosomes simultaneously move on the same mRNA track each polymerizing one copy of the same protein. This phenomenon is usually referred to as ribosome traffic because of its superficial similarity with vehicular traffic on highways |
macdonald68 macdonald69 lakatos03 shaw03 shaw04a shaw04b chou03 chou04 dong1 dong2 cook ciandrini basu07 mitarai08 In this paper we also report the effect of steric interactions of the ribosomes in ribosome traffic on their DTD. |
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