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[EQUATION] where we have used [MATH] In the limit of long time series, the sums self-average and become: [EQUATION] [EQUATION] [EQUATION] |
which yields [MATH] and [MATH] using Eqs. 51 and 52 To obtain the uncertainties, we calculate the second derivatives of the log-likelihood and use the Cramér-Rao bound. After a calculation similar to Eqs. 35 37 , we find: |
[EQUATION] Therefore, maximum likelihood estimation is equivalent to linear regression in the limit of long time series. (One can similarly show that the same conclusion applies to the sensing of spatial gradients: the estimate from linear regression derived in Endres and Wingreen ( 2008 is equivalent to maximum likeli... |
A.3 Monitoring sphere Linear regression Following Berg and Purcell Berg and Purcell ( 1977 , we now consider a perfect monitoring sphere of radius [MATH] , which lets particles freely diffuse in and out of the sphere. This device records the number of particles [MATH] inside the sphere at all times, with ensemble avera... |
[EQUATION] where [MATH] is the volume of the sphere. In the limit of long times, this estimate gives the true value of the concentration and the ramp rate: |
[MATH] [MATH] . The uncertainties are given by the variances of [MATH] and [MATH] . For example, the uncertainty of the concentration estimate is: |
[EQUATION] where [MATH] To compute [MATH] , we decompose [MATH] as a sum over [MATH] independent particles in a large volume [MATH] [MATH] , where [MATH] is a binary variable for each particle, whose value is [MATH] if the particle is in the sphere, and [MATH] otherwise. We assume that [MATH] and [MATH] are large, such... |
[EQUATION] where [MATH] the binary indicator variable of a single particle, and [MATH] . In the limit of large time differences [MATH] , the autocorrelation function [MATH] decays to [MATH] . Since [MATH] , Eq. 61 simplifies to: |
[EQUATION] where we have used time translation invariance and time-reversal symmetry between [MATH] and [MATH] . The quantity [MATH] is the probability that a particular particle was in the sphere at times [MATH] and [MATH] i.e. |
[MATH] , with [MATH] and [EQUATION] where the integrals are over the sphere of radius [MATH] A calculation familiar from electrostatic done in Berg and Purcell ( 1977 yields: |
[EQUATION] Combining the above results and using [MATH] , we obtain the Berg and Purcell bound: [EQUATION] The calculation of the uncertainty of the ramp rate proceeds very similarly: |
[EQUATION] where we have used [MATH] . This approximation is justified by the fact that for [MATH] [MATH] Finally we obtain: [EQUATION] |
Appendix B Uncertainties in a ramp-sensing biochemical network B.1 Input noise We first consider the most general case of the biochemical network shown in Fig. 2 of the main text. Ligands bind the receptor at a rate [MATH] . When a ligand binds to the receptor, it remains bound for a time [MATH] , and [MATH] signaling ... |
[EQUATION] where, in the first equation, [MATH] when the receptor is bound and [MATH] otherwise. When there is only one bound state, [MATH] has an exponential distribution, with [MATH] and [MATH] For averaging times much longer than the durations of bound and unbound intervals, [MATH] can be approximated by a Gaussian ... |
[EQUATION] To estimate the variance of [MATH] for fixed-size burst signaling, we calculate the variance of the number of binding events during a time [MATH] (but [MATH] ). The variance of the duration of one binding/unbinding cycle is: [MATH] . If there are [MATH] binding/unbinding events during [MATH] , the relative v... |
[EQUATION] hence: [EQUATION] Thus, for averaging times much longer than [MATH] and [MATH] , we may write: [EQUATION] Similarly, we can estimate the fluctuations of [MATH] for continuous signaling: |
[EQUATION] Setting [MATH] and [MATH] without loss of generality, we obtain for both cases: [EQUATION] with [MATH] for continous signaling, and [MATH] for fixed-size burst signaling. Note that for an exponential distribution of bound intervals durations [MATH] , so that [MATH] for continuous signaling, and [MATH] for fi... |
[EQUATION] with [MATH] for continuous signaling and [MATH] for fixed-size burst signaling. B.2 Output noise Consider the network shown in Fig. 2 of the main text and described by the equations: |
[EQUATION] When the network is presented with a slow concentration ramp [MATH] (such that [MATH] ), with the input [MATH] given by Eq. 74 i.e. |
[MATH] with: [EQUATION] the average network response is: [EQUATION] with [MATH] We collect the fluctuations of [MATH] and [MATH] [MATH] and [MATH] , into a single vector: |
[EQUATION] Linearizing Eqs. ( 75 ) and ( 76 ) yields: [EQUATION] Multiplying by [MATH] on both sides and integrating, one obtains: |
[EQUATION] with [EQUATION] [MATH] if [MATH] , and the same expression with [MATH] [MATH] [MATH] , otherwise. The fluctuations of [MATH] are given by: |
[EQUATION] and the average uncertainty: [EQUATION] where the last equality is valid at steady state. Exactly the same result is obtained when [MATH] Using Eqs. 78 and 80 , we derive the uncertainty of the ramp rate readout: |
[EQUATION] In addition we can also evaluate the variance of [MATH] [EQUATION] and therefore: [EQUATION] from which we obtain the uncertainty of the concentration readout, using Eqs. 77 and 82 |
[EQUATION] Appendix C Distribution of intervals of signaling activity for a receptor at equilibrium We now prove that for any receptor in thermal equilibrium: |
[EQUATION] where [MATH] is the duration of an interval of uninterrupted signaling activity. With the results from the previous section, this proves [MATH] for any continuously signaling receptor, unless free energy is consumed in the binding/unbinding cycle. Previously and in the main text, the receptor was assumed to ... |
Our goal is to calculate the distribution of intervals of uninterrupted activity when the system is in equilibrium, i.e. intervals where the system remains in [MATH] . In particular, we will show that this distribution is a weighted sum of exponentials with positive weights: |
[EQUATION] This result generalizes the one proven in the supporting information of Tu ( 2008 . In that proof, active states were paired one-to-one to inactive states (whence [MATH] ), and, among the possible transitions from active to inactive states, only the ones within these pairs were allowed. Our proof places no r... |
A consequence of Eq. 93 is that the relative variance of intervals is bounded below by one. This lower bound is only attained for a single-step process ( [MATH] ) for which the distribution of intervals is a pure exponential. More precisely, |
[EQUATION] where the last step follows from Jensen’s inequality, which becomes an equality if and only if [MATH] We now prove Eq. 93 Let [MATH] denote the matrix of rates from state [MATH] to state [MATH] , with [MATH] Call [MATH] the vector of equilibrium probabilities, which satisfies [MATH] Because of detailed balan... |
[EQUATION] Note that in Tu ( 2008 [MATH] and [MATH] are square, diagonal matrices. Assuming that the system enters an active state at time [MATH] , this state will be [MATH] with probability |
[EQUATION] where [MATH] is the submatrix of rates from inactive to active states, [MATH] is the projection of [MATH] onto subset [MATH] , and [MATH] is a vector of ones of dimension [MATH] . Starting from an active state [MATH] at time [MATH] , the probability of still being active in state [MATH] at time [MATH] (if [M... |
[EQUATION] where [MATH] is the same as [MATH] but with absorbing inactive states: [MATH] and [MATH] . Thus: [EQUATION] Therefore the total probability of having exited the active states at time [MATH] , after starting in [MATH] at [MATH] , is: |
[EQUATION] where [MATH] is a vector of ones of dimension [MATH] . Finally, the total probability of having exited the active state at time [MATH] after entering it at time [MATH] is: |
[EQUATION] which yields the probability distribution of intervals of activity: [EQUATION] To exploit the property of detailed balance, we symmetrize [MATH] by defining [MATH] [MATH] being symmetric, it can diagonalized in an orthonormal base: |
[EQUATION] The diagonal form of [MATH] thus reads: [EQUATION] where [MATH] and [MATH] are the right and left eigenvectors of [MATH] [MATH] can now be rewritten as: [EQUATION] We have: [EQUATION] and therefore: [EQUATION] which establishes Eq. 93 with [MATH] |
# Source: arxiv 1003.0814 # Title: Origin of broad polydispersion in functionalized dendrimers and its effects on cancer cell binding affinity # Sections: all # Downloaded: 2026-03-03T05:15:05.409919+00:00 |
Origin of broad polydispersion in functionalized dendrimers and its effects on cancer cell binding affinity Abstract Nanoparticles with multiple ligands have been proposed for use in nanomedicine. The multiple targeting ligands on each nanoparticle can bind to several locations on a cell surface facilitating both drug ... |
A dendrimer is a branched polymeric nanoparticle with the topology of a Cayley tree Dykes ( 2001 ; see Figure . We will be concerned here with dendrimers with a radius of about 5 nm having [MATH] termini that can be functionalized by the conjugation of various endgroups and ligands. These terminal groups can be varied ... |
Venuganti and Perumal ( 2008 Targeting ligands can be used to enhance the binding of the nanoparticle to specific receptors Stella et al. ( 2000 . For example, epithieal cancer cells are known to overexpress folic acid receptors, so that folic acid attached to the dendrimer should target epithelial cancer, allowing che... |
In our experiments measuring the distribution of ligand attachment Mullen et al. ( 2008 we conjugated varying amounts of the ligand 3-(4-(prop-2-ynyloxy)phenyl)propanoic acid to the surface primary amines of a poly(amidoamine) dendrimer (G5 PAMAM; [MATH] . This ligand was chosen because its binding properties and steri... |
The dendrimer was experimentally determined to have approximately 110 free sites at the end of its (roughly) spherically arranged branches. This is modeled as a 11x11 triangular lattice with periodic boundary conditions. We use a kinetic model of cooperative ligation with two parallel attachment paths. In the free atta... |
For large [MATH] , two factors must be considered. First, when a reaction site is proximate to a previously ligated site there is a possibility for a catalytic enhancement of the reaction rate. The reaction occurs at the amide group of the ligand-nanoparticle bond, so the presence of multiple neighbors can increase the... |
[EQUATION] Here [MATH] is a molecular time scale, [MATH] is the free ligand concentration, and [MATH] is a steric hindrance term that is equal to one if [MATH] and is zero otherwise. |
We implemented a continuous-time, rejection-free Monte-Carlo simulation of the model with [MATH] dendrimers. The reciprocal of [MATH] is taken as the time unit. The only remaining variable is the free energy, [MATH] Simulations were run for values of average ligand number [MATH] , resulting in a best fit value of [MATH... |
We also used dendrimers with an average of 68% of the active sites blocked by the conjugation of acetamide groups (G5 PAMAM; [MATH] . We represented this by first using the model above to add acetamide groups using the acetamide-acetamide catalytic interaction free energy barrier [MATH] for [MATH] in eqn (though withou... |
[EQUATION] Because the catalyzing amide bond is present in both the acetamide- and the ligand-nanoparticle bond, we set [MATH] . We have no new parameters, but still fit the data quite well with one parameter; see Figure |
Having undersood the distribution of ligands on the nanoparticle we now turn to their adsorption and desorption from a protein-modified surface. Surface Plasmon Resonance (SPR) can sensitively detect the amount of material adsorbed onto the surface. In Hong et al. ( 2007 , SPR was used to determine the amount of folic ... |
Multivalency, i.e. how multiple bonds between ligands on the nanoparticle and receptors on the surface effect binding is, in general, very complex. However, if [MATH] is broad, apparent multivalent behavior may, in fact, be due simply to fluctuations in ligand number, as we will show. We explain the data of Hong et al.... |
This changes the analysis of the SPR data from that given in Hong et al. ( 2007 . In the standard multivalent model with nanoparticles with different numbers of bonds to the surface, we expect the dissociation to involve many different rates. However, if multiply bound dendrimers do not desorb over the course of the ex... |
The number of singly bound nanoparticles is the fraction of bindable nanoparticles (i.e. with [MATH] ) that have exactly one ligand. Recall that if [MATH] is Poisson then [MATH] . However, since unligated nanoparticles are not bound at all, the fraction of bindable nanoparticles with one FA is [MATH] . This simple mode... |
As above, we use kinetic Monte-Carlo to find [MATH] for FA on the dendrimers. From this, we estimate the fraction of singly-ligated nanoparticles, and thus what fraction of the material we expect to remain bound on the SPR surface. The results are shown in figure . The SPR data is consistent with an approximately const... |
There are theories of multivalent interactions in monodisperse systems in the literature Huskens et al. ( 2004 ); Diestler and Knapp ( 2008 . As we have pointed out, this sort of treatment is not necessary if time-scale separation exists between single-ligand interactions and multiple-ligand interactions. This is the c... |
Monodisperse systems for which multivalent theories are posited may be rarer than anticipated; this is particularly the case for those systems in which dispersion is meant to be controlled by restricting the free ligand concentration (such as in Mullen et al. ( 2008 ); Ackerson et al. ( 2006 ); Huo and Worden ( 2007 ).... |
Our goal in this paper is twofold: first we present an interesting mesoscopic system with its microscopic and macroscopic characteristics completely described by a simple statistical model. Also, we demonstrate that multivalent binding behavior observed in these and related chemical systems need not be explained by exo... |
The model suggests that the strong steric hindrance prohibits newly attached ligands having more than one neighbor. We have examined our simulation results, and we find that this leads to a much larger than chance occurrence of isolated pairs of ligands and, at larger [MATH] , linear arrangements of ligands. If true, t... |
Acknowledgements. This project has been funded in part with Federal funds from the National Cancer Institute, National Institutes of Health, under Award 1 R01 CA119409, and with Federal funds from the National Science Foundation under Award DMS 0554587. |
# Source: arxiv 1003.4628 # Title: Efficient Construction, Update and Downdate Of The Coefficients Of Interpolants Based On Polynomials Satisfying A Three-Term Recurrence Relation # Sections: all # Downloaded: 2026-03-03T02:20:37.691449+00:00 |
Efficient Construction, Update and Downdate Of The Coefficients Of Interpolants Based On Polynomials Satisfying A Three-Term Recurrence Relation |
Abstract In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the interpolation incrementally and can be used to update the coefficients wh... |
Introduction In many applications, we are interested in computing the coefficients of a polynomial interpolation of discrete data relative to a basis of polynomials [MATH] of increasing degree [MATH] In practical terms, this means that given [MATH] function values [MATH] at the |
[MATH] nodes [MATH] [MATH] , we want to compute the coefficients [MATH] [MATH] of a polynomial [MATH] of degree [MATH] such that |
[EQUATION] That is, the polynomial [MATH] interpolates the [MATH] function values [MATH] at the nodes [MATH] If we are only interested in evaluating [MATH] at different [MATH] then the method of choice is Barycentric Lagrange Interpolation |
ref:Berrut2004 , which avoids representing the interpolant in any specific base. Such coefficient-based representations are usefull, however, if we are interested in computing other quantities such as the integral or derivative of the interpolant, its [MATH] -norm and/or performing other operations on it such as transf... |
In the following, we will assume that the polynomials [MATH] of degree [MATH] can be constructed using a three-term recurrence relation, which we will write as |
[EQUATION] with [EQUATION] Examples of such polynomials are the Legendre polynomials [MATH] with [EQUATION] or the Chebyshev polynomials [MATH] with |
[EQUATION] The coefficients [MATH] of Equation ( ) can be computed solving the system of linear equations [EQUATION] which can be written as |
[EQUATION] The matrix [MATH] is a Vandermonde-like matrix and the system of equations can be solved in [MATH] using Gaussian elimination. As with the computation of the monomial coefficients, the matrix may be ill-conditioned ref:Gautschi1983 |
\citeasnoun ref:Bjorck1970 present an algorithm to compute the monomial coefficients of an interpolation without the expensive and potentially unstable solution of a Vandermonde system using Gaussian elimination, by computing first the coefficients of a Newton interpolation and then converting these to monomial coeffic... |
[MATH] [MATH] is used. In Section , we will re-formulate the algorithms of Björck and Pereyra and of Higham and extend them to update the coefficients after a downdate i.e. the removal of a node, of an interpolation. In Section we present a new algorithm for the construction of interpolations of the type of Equation ( ... |
A Modification of Björck and Pereyra’s and of Higham’s Algorithms Allowing Downdates \citeasnoun ref:Bjorck1970 present an algorithm which exploits the recursive definition of the Newton polynomials |
[EQUATION] They note that given the Newton interpolation coefficients [MATH] the interpolation polynomial can be constructed using Horner’s scheme: |
[EQUATION] where the interpolation polynomial is [MATH] They also note that given a monomial representation for [MATH] such as [EQUATION] |
then the polynomial [MATH] can be constructed, following the recursion in Equation ( ), as [EQUATION] From Equation ( ) we can then extract the new coefficients [MATH] |
[EQUATION] \citeasnoun ref:Higham1988 uses the same approach, yet represents the Newton polynomials as a linear combination of polynomials satisfying a three-term recurrence relation. Using such a representation |
[EQUATION] he computes [MATH] by expanding the recursion in Equation ( using the representation Equation ( 11 ): [EQUATION] Expanding Equation ( 12 ) for the individual [MATH] , and keeping in mind that [MATH] , we obtain |
[EQUATION] By shifting the sums in Equation ( 13 ) and re-grouping around the individual [MATH] we finally obtain [EQUATION] Higham then extracts the new coefficients [MATH] from Equation ( 14 ) as: |
[EQUATION] In both algorithms, the interpolating polynomial is constructed by first computing the divided differences [EQUATION] |
and, starting with [MATH] , and hence [MATH] or [MATH] successively updating the coefficients per Equation ( 15 ) or Equation ( 10 respectively. |
Alternatively, we could use the same approach to compute the coefficients of the Newton polynomials themselves [EQUATION] Expanding the recurrence relation in Equation ( ) analogously to Equation ( 14 ), we get |
[EQUATION] We initialize with [MATH] and use [EQUATION] to compute the coefficients for [MATH] [MATH] Alongside this computation, we can also compute the coefficients of a sequence of polynomials [MATH] of increasing degree [MATH] |
[EQUATION] initializing with [MATH] , where the [MATH] are still the Newton coefficients computed and used above. The subsequent coefficients [MATH] [MATH] are computed using |
[EQUATION] This incremental construction of the coefficients, which is equivalent to effecting the summation of the weighted Newton polynomials and is referred to by Björck and Pereyra as the “progressive algorithm”, can be used to efficiently update an interpolation. If the coefficients [MATH] and [MATH] are stored an... |
Algorithm 1 Incremental construction of [MATH] 1: [MATH] init [MATH] 2: [MATH] [MATH] init [MATH] 3: for [MATH] do 4: [MATH] [MATH] |
init [MATH] 5: for [MATH] do 6: [MATH] compute the [MATH] 7: end for 8: [MATH] compute [MATH] 9: [MATH] compute [MATH] 10: [MATH] |
compute [MATH] , Equation ( 20 11: [MATH] compute the new [MATH] , Equation ( 19 12: [MATH] [MATH] compute the new [MATH] , Equation ( 18 |
13: end for We can re-write the recursion for the coefficients [MATH] of the Newton polynomials in matrix-vector notation as [EQUATION] |
where [MATH] is the [MATH] tri-diagonal matrix [EQUATION] and [MATH] is a [MATH] matrix with [MATH] in the diagonal and zeros elsewhere |
[EQUATION] The vectors [MATH] and [MATH] contain the coefficients of the [MATH] th and [MATH] st Newton polynomial respectively. |
Given the vector of coefficients [MATH] of an interpolation polynomial [MATH] of degree [MATH] and the vector of coefficients [MATH] of the [MATH] st Newton polynomial over the [MATH] nodes, we can update the interpolation for a new node [MATH] and function value [MATH] |
as follows: Instead of computing the new Newton interpolation coefficient [MATH] using the divided differences as in Equation ( 16 ), we choose [MATH] such that the new interpolation constraint |
[EQUATION] is satisfied, resulting in [EQUATION] which can be computed by evaluating [MATH] and [MATH] Note that since [MATH] for [MATH] , the addition of any multiple of [MATH] to [MATH] does not affect the interpolation at the other nodes at all. This expression for [MATH] is used instead of the divided difference si... |
We then update the coefficients of the interpolating polynomial using [EQUATION] and then the coefficients of the Newton polynomial using |
[EQUATION] such that it is ready for further updates. Starting with [MATH] and [MATH] , this update can be used to construct [MATH] by adding each [MATH] and [MATH] |
[MATH] , successively. The complete algorithm doing just that is shown in Algorithm The addition of each [MATH] th node requires [MATH] operations, resulting in a total of [MATH] operations for the construction of an [MATH] -node interpolation. |
This is essentially the progressive algorithm of Björck and Pereyra, yet instead of storing the Newton coefficients [MATH] we store the coefficients [MATH] of the last Newton polynomial. This new representation offers no obvious advantage for the update, other than that it can be easily reversed |
Given an interpolation over a set of [MATH] nodes [MATH] and function values [MATH] [MATH] defined by the coefficients [MATH] and given the coefficients [MATH] of the [MATH] Newton polynomial over the same nodes, we will downdate the interpolation by removing the function value [MATH] at the node [MATH] The resulting p... |
We start by removing the root [MATH] from the [MATH] st Newton polynomial by solving [EQUATION] for the vector of coefficients [MATH] Since [MATH] is a root of [MATH] , the system is over-determined yet has a unique solution We can therefore remove the first row of [MATH] |
and the first entry of [MATH] , resulting in the upper-tridiagonal system of linear equations [EQUATION] which can be conveniently solved in [MATH] using back-substitution. |
Once we have our downdated [MATH] , and thus the downdated Newton polynomial [MATH] , we can downdate the coefficients of [MATH] by computing |
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