Gu and a transition temperature Tm. For a long time, it has been known that solutes can change Tm profoundly. Here we present a phenomenological model that describes the change of Tm with the solute concentration cs in terms of two effects: (i) the change of the number of correlated counterions Δnci and (ii) the change of hydration expressed through the parameter Δw and its dependence on temperature expressed through the parameter dΔcp/dcs. Proteins always carry charges and Δnci describes the uptake or release of counterions during the transition. Likewise, the parameter Δw measures the uptake or release of water during the transition. The transition takes place in a reservoir with a given salt concentration cs that defines also the activity of water. The parameter Δnci is a measure for the gain or loss of free energy because of the release or uptake of ions and is related to purely entropic effects that scale with ln cs. Δw describes the effect on ΔGu through the loss or uptake of water molecules and contains enthalpic as well as entropic effects that scale with cs. It is related to the enthalpy of transition ΔHu through a Maxwell relation: the dependence of ΔHu on cs is proportional to the dependence of Δw on temperature. While ionic effects embodied in Δnci are independent of the kind of salt, the hydration effects described through Δw are directly related to Hofmeister effects of the various salt ions. A comparison with literature data underscores the general validity of the model.The unfolding transition of proteins in aqueous solution containing various salts or uncharged solutes is a classical subject of biophysics. In many cases, this transition is a well-defined two-stage equilibrium process which can be described by a free energy of transition Δ A phenomenological approach to the unfolding transition of proteins is given. The model treats quantitatively the effect of electrostatics as well as of hydration (Hofmeister effects). Cold denaturation,2 which has been known for a long time, is the transition to denatured state taking place with decreasing temperature. It is well-established that for many proteins chain denaturation is a two state transition3–6 in which the globular and the denatured form of the protein are well-defined thermodynamic states in equilibrium with each other. Hence, an equilibrium constant Ku can be defined between the globular and denatured state which allows us to treat the denaturation as a fully thermodynamic problem relating the melting temperature Tm to the transition enthalpy ΔHu and the transition entropy ΔSu.7The denaturation of proteins by a globule to coil transition is a classical subject of biophysics.Tm of a given protein with solutes in the aqueous phase. Up to now, there have been an enormous number of experimental studies that started out in the sixties of the last century.8 There are many investigations that study the change of Tm in the presence of various salts and non-charged solutes which can stabilize or destabilize the globular state.4,6,7,9–20 This effect is of obvious biological importance and can be traced back to hydration effects embodied in the Hofmeister series.21–24 The collapse transition of poly(n-isopropylacrylamide) (PNIPAM) in aqueous solution is another well-studied and fundamental problem where a coiled polymer undergoes a transition from the coiled to the globular state with raising temperature. Here too there is a large number of fundamental and detailed studies on this transition in solutions of various ions.23,25–30 Taken together the folding/unfolding transition of proteins and polymers in general is problem of fundamental importance.A fundamental problem in the field is the change 1 The unfolding of the globular protein exposes charged groups to water and this interaction leads to an important contribution to the free energy of unfolding that scales with the logarithm of the salt concentration in solution.1 This term is due to the release or uptake of ions during unfolding and play an important role both for unfolding of proteins as well as for denaturing of DNA in presence of various salts .34 Temperature T and salt concentration cs were identified as the decisive variables and a closed expression for the free energy ΔGb of complex formation could be derived. In this model counterion release was characterized by Δnci denoting the net number of released ions during binding whereas hydration was described in terms of the parameter Δw defined already in early expositions of the problem1,43,44 and used frequently to describe the effect of hydration on complex formation.44–49 Central for the development of this model is the fact that mixed derivatives of the binding enthalpy ΔHb with regard to T and cs must be the same. Hence, this Maxwell-relation leads to prediction that the dependence of ΔHb on cs gives directly the dependence of Δw on temperature. The model thus derived is capable of describing the weak dependence of ΔGb on temperature which in turn leads to a strong compensation of enthalpy and entropy.34 Moreover, the values obtained for Δnci and Δw obtained by the present model for the denaturation of a given protein can directly be compared to data deriving from studies of complex formation of polyelectrolytes with proteins.33,46,47,50,51Surveying the literature on denaturation of proteins, it becomes clear that exchange of water and counterions during unfolding present two important factors that determines the stability of proteins in aqueous solution to a large extend. Both are modified by the added solute. Hence, a quantitative treatment of the effect of ions and water is a necessary prerequisite for a quantitative evaluation of data related to the unfolding of proteins in presence of various solutes. In a recent paper we have presented a unified approach for the free energy of complex formation between proteins and polyelectrolytes that comprises both effects.16,18 The entire discussion presented here aims at a systematic analysis of experimental data obtained on polymeric unfolding transitions of various systems in aqueous phase.Based on this model we here present a phenomenological approach to unfolding transitions of proteins that are partially charged. A closed expression for the free energy of unfolding will be presented that contains both the effect of electrostatics as well as of hydration. The consequences of the model for data evaluation will be discussed and exemplified using recent experimental data.e.g. by measurements of the circular dichroism leading to a fraction α of unfolded protein. The equilibrium constant Ku for the process of unfolding is related to α byGu is related to Ku throughWe consider the transition of a single chain of a polypeptide from a folded to an unfolded state in sufficiently dilute solution. In each stage of this transition the unfolded state is in equilibrium with the still folded part of the chain. This two-state mechanism is well-established for a great number of systems  the change of the activity coefficients γf and γu with the activity a± of the added salt give a small but non-negligible contribution of the term Δnci cs = ΔHb/RT2 we obtain the differential of ln Ku for monovalent ionsHu denotes the enthalpy change at the unfolding transition and Tm the respective temperature of unfolding. Thus, in the following the unfolding transition will be treated as the function of the two decisive variables, namely temperature Tm and salt concentration cs.Hence, the salt concentration nci is independent of temperature.32,34,35,37,52–54 It is therefore safe to disregard the dependence of this parameter on Tm. With this assumption and34There is abundant experimental evidence that the parameter Δw on temperature. As already lined out previously,34 this relation can now be used to calculate Δw as the function of temperature. In general, the transition enthalpy ΔHu as the function of the melting temperature Tm and cs can be rendered as34cp,0 denotes the change of the specific heat in absence of added salt whereas the coefficient dcp/dcs describes the change of the specific heat with salt or solute concentration.34T0m denotes the melting temperature for salt-free solutions. Together with This relation demonstrates that the salt dependence of transition enthalpy is directly related to the dependence of the parameter Δ34w(T0m) denotes the magnitude of Δw at T0m in salt-free solution.Integration leads to34 Δw can be interpreted in terms of the solute partitioning model as follows. Both the polyelectrolyte as well as the protein are hydrated in aqueous solution. During the unfolding a certain number Δnw of water molecules of both reactants is taken up or released. Furthermore, it is assumed that there is a partitioning of the ions between the bulk solution and the hydration water on the surface of the protein described by the partition coefficient Kp,+ = (mloc+/mbulk+) for the cations where mloc+ denotes the molality of the cations in the hydrated shell whereas mbulk+ is the respective quantity in bulk. The partition coefficient Kp,− of the anions is defined in the same way. With these definitions, Δw can be rendered by34As already discussed previously,w measures the effect of water release on the free energy of unfolding and should not be confused with the total number Δnw taken up or released during unfolding. For an equal distribution of the ions between the hydrate and the bulk phase, this contribution will vanish.Evidently, the quantity Δnci = 0. Integration of K0u is the equilibrium constant in salt-free solution. ThereforeG0u denotes the free energy of unfolding at cs = 0. Hence, the dependence of ΔGu on cs can be written down asIn the following, we first consider uncharged systems, that is, ΔTm − T0m does not exceed 10 degrees so that the last term in In many cases the difference m-value defined as the derivative of the free energy with regard to solute concentration at constant temperaturem is given by a constant plus a term that depends quadratically on Tm − T0m. For small temperature differences the second term will be small and the m-value is a constant in good approximation. However, it should be noted that m is in general a quantity that depends explicitly on temperature.This expression shows that cs. The quantity ΔG0u for salt- or solute-free solutions can be derived following the prescription of Chen and Schellman:6 the specific heat Δcp,0 measured in solute-free systems can be regarded as a constant throughout the rather small temperature range under consideration here. Thus, for solute-free systems we obtainCombination with Tm − T0m ≤ 10 K this expression can be approximated byFor cs must be added to 1 Here it must be kept in mind that there is always a small but finite salt concentration cs,0 so that the integration of nci does not depend on temperature we immediately obtain from eqn (22)For partially charged proteins c0s is small and can be disregarded in c0s the free energy of unfolding may contain an appreciable contribution originating from the release or uptake of ions during denaturation. Hence, ΔGu will be dominated by the last term for small cs. The respective transition enthalpy is given by cs is replaced by cs − cs,0. The transition entropy follows asIn many cases the concentration Gu which gives the change of ΔGu with cs calculated for the transition temperature T0m in solute-free solution:In many cases it is only possible to deduct the change of the free energy of unfolding with increasing solute concentration. Thus, we require the quantity ΔΔ6  which is closely related to the classical m-value through cp,0 in absence of solutes; and (iii) the parameter dcp/dcs describing the dependence of Δcp on cs. This parameter has been introduced by Chen and Schellman as well  Δ29 Here, the hydration effects are described in terms of the preferential interaction parameter ΔΓ23. If this parameter does not depend on cs, it follows directly thatwhere  is defined as the preferential interaction parameter independent of cs. The analysis of the changes effected by kosmotropic salts showed indeed that this equation provides a very good approximation of the experimental data obtained for the collapse transition of PNIPAM-chains in aqueous solution.26,29 Thus, these data could be compared directly to the prediction of the SPM with moderate success  effected by unfolding and a length parameter l (∼ 0.1 nm) describing the thickness of the layer inaccessible for the ions. The estimate of the m-value derived from this calculation compares favorably with the measured values  for all pertinent temperatures ranging from cold to thermal denaturation. Given the fact, however, that Δcp,0 depends on temperature,7 such stability curves should be regarded with caution.In principle, 4,33,34e.g. DNA with various proteins.33,46,47,50,51 It is thus interesting to compare its magnitude for complex formation with values deriving from protein unfolding. Evidently, the parameter Δw introduced by this equation does not give the number of released water molecules defined as Δnw but measures the thermodynamic effect of this release .(1) In a first step of the analysis of experimental data, dependence of Δ as well . If, on cs will profoundly change the dependence of the free energy on salt concentration and this dependence will be most marked for small cs  The term scaling with ln ll cs cf.. The depT, the combination of both expressions shows that in this caseGu/ΔH0u ≈ ΔT/Tm should be an universal function. Hence, for small ΔT, this equation may be used to check the internal consistency of data .(3) If the term quadratic in Hu(cs) at different concentrations of the solute cs and the melting temperature Tm at the respective salt concentration cs  for a typical kosmotropic salt as NaCl as well as for NaSCN which provides a good example for a chaotropic system. The enthalpy of denaturation in presence of NaCl hardly depends on salt concentration whereas a marked dependence is found for NaSCN. This test splits up the experimental data sets into two classes:As outlined above, the analysis may start by the check of the dependence of ΔTm; kosmotropic ions: the small dependence of ΔHu on cs suggests that the coefficient dΔcp/dcs in nci and Δw(T0m). Moreover, the changes ΔT = Tm − T0m are rather small so the term quadratic in ΔT in (1) Small ΔT; chaotropic ions: for NaSCN there is a marked dependence of ΔHu(cs) on salt concentration which immediately demonstrates that dΔcp/dcs assumes a finite value and the m-value  Large Δ m-value  in turn Tm; kosmotropic ions: Hu as the function of the difference Tm − T0m. The error of these numbers is of appreciable magnitude and only allows us to obtain an estimate for Δcp,0 for which an evaluation for the data of all kosmotropic ions  gives an estimate Δcp,0 ≅ 4.6 kJ (K−1 mol−1) which compares well literature (see ref. cp/dcs = 0.Case (1): small Δsee ref.  and 58. Tm as the function of salt concentration obtained by numerical solution of Gu = 0. Here the data Tm(cs) obtained for a given salt are fitted to T using the MathLab routine cftool (MATLAB (2021b). Natick, Massachusetts: The MathWorks Inc.). All calculations have been done using the value of the transition enthalpy in salt-free systems ΔH0u = 392 kJ mol−1 and the transition temperature T0m = 326.8 K given by Francisco et al.18 As mentioned above, the buffer added to all solutions leads to a cs,0 = 0.01 M.18 The solid lines in nci turned out to describe ΔGu for all systems under consideration here in agreement with the above general considerations. This fact has already been observed by Francisco et al.18 and the presence analysis compares well with Tm can be described by the combination of a linear and a logarithmic term  can be compared to data obtained for complex formation of polyelectrolytes with proteins.term see . Pegram nci is positive for all kosmotropic salt analyzed herein. This finding points to the fact that a small but finite number of ions attached closely to the surface of the protein is released during the unfolding transition. With increasing cs these ions are released into a reservoir with increasing activity which requires additional free energy during the unfolding transition. Hence, this effect stabilizes the folded state and leads to a higher transition temperature.The parameter Δw(T0m) is negative which means that the water molecules needed for the hydration of the unfolded protein must have a higher activity as the bulk water since addition of these salts increases the magnitude of ΔGu. Hence, free energy is needed to transport water from a state of lower activity in bulk to a state of higher activity in the hydrate shell upon unfolding of the protein. This effect is due to a partial depletion of these kosmotropic ions from the hydrate shell of the protein and leads to a stabilization of the folded state. The magnitude of Δw(T0m) found here is in the same range as found previously for complex formation of proteins with DNA.47The parameter ΔGu but also ΔHu at the same time. Thus, the independence of the m-value of temperature follows here from an analysis of the latter quantity. Only this analysis allows us to disregard the term in T2.It should be noted that the present analysis not only treats ΔT; chaotropic ions: in the following, the evaluation of the respective parameters will be shown using the data for NaSCN (Table 1 of ref. Hu(cs) as the function of ΔTm whereas the solid lines displays the fit of these data according to Hu(cs = 0) = 392 kJ mol−1 and the specific heat Δcp,0 = 4.6 kJ (K−1 mol−1) estimated from the analysis of the kosmotropic systems shown in cp/dcs a value of ca. 2.5 kJ (K−1 mol−1 M−1). Evidently, the small range of data and the finite accuracy of the data allows for an estimate of these quantities only. However, since these parameters present only corrections in Case (2): large Δcp,0 = 4.6 kJ (K−1 mol−1) and dΔcp/dcs = 2.5 kJ (K−1 mol−1 M−1) are introduced into nci and Δw(T0m) are derived from a numerical solution of this equation for ΔGu = 0. Input parameters are the measured Tm measured for different NaSCN-concentrations marked by points in Tm calculated with the parameters Δcp,0 = 4.6 kJ (K−1 mol−1), dΔcp/dcs = 2.5 kJ (K−1 mol−1 M−1) and the values of Δnci and Δw(T0m). Again, a full description of the experimental transition temperatures is achieved. For the chaotropic salt NaSCN the parameter Δw(T0m) assumes a positive value which is directly related to the fact that SCN−-ions are adsorbed on the unfolded protein chain thus lowering the activity of the hydrate water molecules. Hence, free energy is gained when hydrating the unfolded chain by bulk water having a higher activity. The parameter Δnci now has assumed a negative value. This finding points to a much stronger interaction of such chaotropic ions with the unfolded protein chains. Thus, Fang and Furo could show that chaotropic ions can associate to PNIPAM chains with a Langmuir-type association behavior while NaCl is only weakly adsorbed.59 This effect measured through careful measurements of the electrophoretic mobility was found strongest for SCN−-ions. Hence, adsorption of chaotropic ions can diminish or even reverse the effective charge of unfolded proteins. However, further investigations of Tm at very low ion concentrations are needed to clarify this problem.In the next step, the parameters Δnci with a negative Δw value should lead to a non-monotonic dependence of Tm on salt concentration. This effect is seen in a careful study of the unfolding of RNase A in NaCl solutions by Senske et al.16 These data have been taken using a 50 mM citrate buffer at pH = 5 and are hence not directly comparable to the data of Francisco et al.18 discussed above. T in nci = −0.60 and Δw = −31.7. At small salt concentrations, the logarithmic term in Tm again.As mentioned above, the combination of a negative ΔTm with the solute concentration cs is captured by two effects: (i) the change of the number of correlated counterions Δnci during the unfolding transition, and (ii) the change of hydration expressed through the parameter Δw. The latter parameter is not directly the number of water molecules released or taken up during transition but described the change of the free energy by the release or uptake of water (see the discussion of cs. The enthalpy ΔHu can directly be related to the parameter Δw by the Maxwell-relation cp/dcs describes the direct dependence of ΔHu on salt concentration. The model allows us to discuss the classical m-value in terms of these parameters (m is depending on temperature if the parameter dΔcp/dcs assumes a finite value. A first comparison with experimental data taken from literature shows the general validity of the model.A phenomenological model describing the unfolding transition of proteins has been presented. Within this model, the change of rameters  and predThere are no conflicts to declare.


We minimize a linear combination of the length and the  N regular curves elastic energy functionals the arclength parameter and A regular network The elastic energy functional has a long history. Already at the times of Galileo scientists tried to model elastic rods and strings, looking for equations for equilibrium of moments and forces. The idea to relate the curvature of the fiber of the beam to the bending moment came only later when, in 1691, Jacob Bernoulli proposed to model the bending energy of thin inextensible elastic rods with a functional involving the curvature. Several authors refer to the functional  as EulerWe are interested in the minimization of the functional Notice that the length mentclass2pt{minimOur first task is therefore to describe the class of limits of sequences (equibounded in energy) of regular networks. In other words, we have to characterize the “closure in energy” of the class Then, since the lack of compactness implies that it could not be possible to solve the original minimization problem among regular networks, we relax it by considering the lower semicontinuous envelope of the functional In Propositions weak notion of curvature is well defined, varifolds for instance. The characterization of the relaxation of the original functional in such cases turned out to be particularly difficult  to singular sets for which at least a instance , 16, 19)means of . An advaTo place our paper in the broader context, we mention the minimization of Willmore-type functionals among both Riemannian manifolds and singular structures is an extreme flourishing research field. As the simplest possible example of singular structure we mention surfaces with boundary The main reason for which we got interested in this problem is a previous study of its dynamical counterpart see , 14). Th. Th14]).After this detour on the literature, we explain more in detail what can happen along sequences of regular networks A Model Problem: Theta-NetworksA Theta-network is a regular 3-network whose curves form equal angles at the two junctions Fig. . The limG.The general case presents new interesting features with respect to the model problem of Theta-networks. Firstly, one realizes that much more than the length of one single curve can go to zero, since entire parts of the network can vanish as The Underlying GraphG is composed of edges G captures the topology of the networks and allows us to have a reminiscence of their structure even when some curves in a sequence of networks The oriented graph A Necessary Angle Conditionsingular if it is a constant map, and regular if it is an immersion of class We say that a curve Outer tangents) Consider a regular curve of a network  A (possibly) singular network  A subgraph H of the underlying graph G is stratified-straight if there exists a finite sequence of subgraphs, called strata,the (sub)network the stratum  Let V. An N-graph G is the topological quotient space of E induced by G  is a pair G is an N-graph. Moreover we ask each singular curve) or an immersion of class regular curve).An G the curve We will identify two networks Convergence of networks) We define that a sequence of networks n and each sequence of curves (Singular part) Given a network singular part(p with p belongs to If Regular and singular networks) We say that a network regular if singular otherwise. Consider an N-graph G and p is a junction of order m (with (Graph with assigned angles) We say that an N-graph G has assigned angles if to every junction directionsm is the order of the junction and where(Angle condition for a regular network) Given G an N-graph with assigned angles, we say that the regular network m, writingp such that Consider a (possibly singular) network  is a positive basis of (Gauss–Bonnet-type theorems for curves in the plane) Let mentclass2pt{minimBelow we will need a refined version of this theorem. Suppose now that the curve We discuss here more in detail the example, anticipated in the Introduction, of a singular network that satisfies the angle condition CC. Observe that there exists a choice of virtual tangents for which Let the dimension of the ambient space be entclass1pt{minimaThe angle condition for a singular network is thus not sufficient to characterize the set of limit networks with bounded energy.Straight graph) An N-graph G with assigned angles is straight if there exists a regular network  Given a graph G with assigned angles we say that a subgraph strata,the (sub)network We call step of G the least q for which the above holds.\sim 0,, \,\,  edges with angles G has the structure of a tree with no cycles. In Sect. Notice that the notion of straight and stratified straight graphs coincide when the underlying graph DegenerateN-network) Let G be an N-graph with assigned angles. A network it satisfies the angle condition in the sense of Definition its singular part (We denote by fined in .(Compactness) Let Up to subsequence and up to relabeling the edges n there exists a family of unit vectors i for which Since all the networks of the sequence Define First of all we claim that the angle condition is verified for t verify . It remaC. Hence we can repeat the previous reasoning to conclude that up to subsequence the networks It remains to verify that i≥1, see  below), property  we thus After a finite number of steps the process stops and we obtain that the strata We have thus concluded that To convince the reader that in Definition G where in particularv, w in the plane  For every network To prove the above proposition we will take advantage of the following preliminary lemmas.We recall the following version of a classical lemma.Let f is continuous. In this case, being defined on a compact set, f is uniformly continuous with some modulus of continuity By the triangle inequality and the density of continuous functions on Let l, recalling  Consider two parallel horizontal straight lines in h and the base b are related to h we can choose We construct the curve by putting together two congruent arcs of circle of radius 1 as in Fig. The following lemma allows us to suppose, in the construction of the recovery sequence, that each regular curve of a] its image is a straight segment of length at most (Straightening at vertices) Consider a curve C that depends on Without loss of generality we suppose that We now come to the proof of the recovery sequence.H is connected and G, such that H is stratified-straight with strataConsider H. For every p the image of the curve Fix H, that is we replace in a neighborhood of 0 the completely singular subnetwork We consider the first stratum l. In particular Consider a curve We repeat this process for every curve in Since we can do this for any The functional The existence of minimizers of the functional N-networks such that N-networks and this concludes also the first part of the statement. By definition weakly to strongly to F coincides with the The choice of the weak topology in the above relaxation result is the natural one because of the compactness result of Proposition Now that we have shown that the relaxed Problem G is a 3-graph composed of edges We call Theta-network any planar network The minimization of the elastic energy among Theta-networks has been considered in . It turnG and of the angle condition minimizers are actually in G this could not be the case and so it is hopeless to give always a positive answer to Problem Although we have proven that in general sequences of regular networks with equibounded energy converge to a limit network in G with assigned angles composed of (The minimizers of the relaxed problem are degenerate) Consider the 3-graph G with assigned angles. We recall that We have just seen in Example Let Moreover suppose that for every Denote by We pass now to prove the second part of the statement. Using Cauchy–Schwarz inequality we getG is stratified-straight.We have that fined in  equals z(i)p the convex hull of the outer tangents contains the origin in its interior, that is at every vertex (ii)two edges then Moreover, suppose that one of the following conditions holds: G is stratified straight, there exists a degenerate immersion Let us prove the first claim. Suppose first that G graph with assigned angles encoded by Consider any regular network H is a closed half plane containing p, call it H and whose smallest angle with Suppose now that i) holds. Since the number of vertices is finite there exists at least one junction Suppose now that ii) holds. We then have that the least positive angle As another application of Lemma Let In order to prove the required lower bound we can assume that the curves i there exists We denote by i there exists j such that We are left to show how to prove the claim, namely that for any mentclass2pt{minimN-graph G with assigned angles G we assign an abstract “total turning angle” C  entirely contained in Although the definition of the class of degenerate networks given in Definition From now on we fix the dimension of the ambient space to Path and cycle) Let G be an N-graph with assigned angles. A path composed of cycle if open path. and the last is  sometimes we will simply say that When the first element of a path Let Angle of a path) Let G be an N-graph with assigned angles  Let G be an N-graph with assigned angles. We say that a network (i)its regular curves satisfy the angle condition in the sense of Definition (ii)if (iii)if We call step of G the minimal q for which the above holds.Fix an G be an N-graph with assigned angles. A network it satisfies the angle condition in the sense of Definition the singular part Let G be a graph with assigned angles. Suppose that the ambient space is Let H. We define p and Suppose that ows that  is satisp using rotations of p.Let τ0,1. By  it folloτ0,1. By  is satisNotice that if Now if H, getting that Definition H of Therefore it follows that we can then iterate the above arguments, possibly considering cycles passing through If instead it occurs that Conversely, suppose now that Definition We underline the fact that in Definition G be a graph with assigned angles. Suppose that the ambient space is Let As we already mentioned, we remark again that it is somehow easier to use Definition In this section we study a simple but remarkable case in which we completely characterize stratified straight or straight subgraphs. This helps us to better understand the algebraic and combinatorial relation between these two concepts, together with providing a non-trivial case in which the two definitions are not equivalent.In the whole section we study networks in N-graph G with junctions of order at most four and suppose that for every junction Throughout the section we will consider an H a stratified straight subgraph of G composed of edges a, b, such that, if a or b. In particular, up to rotation, we can assume that Let H a stratified straight subgraph of G composed of the edges i it holdsi and uniquely defines the z, i) and using the rule  Putting t}β=0 in , then \dN-graph G with assigned angles we consider the minimization problemGiven an It is easy to prove that Problem Consider a minimizing sequence of networks e bounds , 3.4) a a\documee bounds  togetherNotice that fixing the length of each curve avoids any form of degeneracy of the limit networks and makes the question on existence of minimizers trivial.


The novel methodology presented here, utilizes a completely free and open source software toolchain for automatically screening unknown alternative foods for toxicity using experimental data from ultra-high-pressure liquid chromatography and mass spectrometry. The process uses three distinct tools  enabling it to be modular and easily upgradable in the future. MZmine 2 and MFAssignR have been previously described, while ToxAssign was developed here to match the formulas output by formula assignment to potentially toxic compounds in a local table, then look up toxic data on the Open Food Tox Database for the matched compounds. This process is designed to fill the gap between food safety analysis techniques and developing alternative food production techniques to allow for new methods of food production to be preliminarily tested before animal testing. The methodology was validated against a previous method using proprietary commercial software. The new process identifies all of the toxic elements the previous process identified with more detailed information than the previous process was able to provide automatically.  Specifications table Agricultural-loss-based global catastrophic risks (GCRs) have some of the greatest probabilities and impacts Leaf concentrate is prepared following standard procedures This analysis method, which differs substantially from the original analysis by Pearce et al. as outlined in The first section of this analysis relies on the software, outlined in the paper by Pluskal et al. Raw Data Import:Raw Data Methods → Raw Data Import as shown in Raw data files.To import a raw data file click on After the raw data is imported, it can be helpful to get an initial view of the chromatogram to determine the retention time where peaks are detected. To do so, hover the mouse over the desired raw data file on the left window and select show TIC on the dropdown menu as shown in This will bring up a parameter menu as shown in An example of an initial chromatogram is shown in Mass detection:Raw data methods → Peak detection → Mass detection as shown in The next step is to run the mass spectra through the peak detection feature to detect the masses from the data. Select the imported files by clicking on them in the left window, then click This will open a window with several parameters to be entered. The next section will outline each parameter section shown on the right-hand side of Mass detector:mass detector and select Exact Mass.First step is to set the mass detector, an algorithm to use for mass detection and its parameters. Click the drop-down box titled Then set the noise level to what matches the data, click the ellipses box to the right of the mass detector drop down box. This will open a parameter window where the noise can be set and previewed. The blue lines in Mass list name:masses as the title.The mass list name is what the name of the detected mass file will be outputted as after the feature is run it will generate a list of points for each scan in the sample. The program auto inputs OK button to run the program. The program progress status is shown in the bottom right side of the window. When mass detection is complete a green check will appear over the file icon next to raw data on the left-hand side of the window. After the process is finished, click on the plus sign box to the left of each data file, this will display a list of the profile spectra. Then click on the plus sign box to the left of each profile spectrum and this will display the centroid spectrum labelled as the mass list name masses. Double click on the masses brings up a window that shows a profile spectrum in blue and the centroid masses in green. The accuracy of the mass detection step can be evaluated by comparing where the centroid spectrum and the profile spectrum line up. The goal is to have the green centroid line intersect the peak of the profile spectra in blue, or if there is no peak, for the green line to run closely parallel to the blue line. An example is shown in Once all parameters are set appropriately for the data set, click the ADAP chromatogram builder:The next step is to use the ADAP chromatogram builder feature, which builds extracted ion chromatograms by taking the mass lists and builds chromatograms for each mass that can be detected continuously over spectrometry scans.Raw data methods → peak detection → ADAP chromatogram builder as shown in To run this feature, click This will display a parameter window for the ADAP chromatogram building. The window of parameters is shown in Scans:Enter MS level 1 and the scan filters will be the same from what was set in the mass detection step of the process, scan parameters are not required unless needed to manipulate the raw data set.Mass list:Enter “masses” or whatever was set as the title for the mass detection results to be used for this programMin group size in # of scans:Minimum scan span over which some peak in the chromatogram must have (continuous) points above the noise level to be recognized as a chromatogram. The chromatography system will determine the optimal value. By evaluating the raw data and it is possible to observe the typical time span of chromatographic peaks and select appropriate values. Typically, a value of 3 to 5 will provide accurate results, 5 will work in most situations. The min group size number of scans detects peaks, so a setting at 3 will detect more peaks but with less confidence than a selection of 5. Selecting 3 can be helpful for small peaks that are eluted very quickly and not observed in many scans.Group intensity threshold:This parameter is the intensity value for which intensities greater than this value can contribute to the minimumScanSpan count. This value should be set to match the noise level  exactly to keep the data set consistent as the chromatogram is being constructed.Min highest intensity:Points below this intensity will not be considered in starting a new chromatogram. The minimum highest intensity should be higher than the noise level. If 200 is entered for noise and intensity threshold, 300–400 would be a good minimum highest intensity, this can be raised to as high as 20,000 based on the minimum intensity used at Michigan Tech in past toxicology projects. Again, this can be adjusted based on results. This parameter should stay consistent between ESI negative and ESI positive datasets. It should be noted that the noise levels can be different between the two ionization modes sometimes, so this may not always be the best option.m/z tolerance:Maximum allowed difference between two m/z values to be considered the same chromatogram. The value is specified both as absolute tolerance (in m/z) and relative tolerance (in ppm). The tolerance range is calculated using maximum of the absolute and relative tolerances. If there is greater confidence in using either m/z or ppm, any value set to 0.0 will not be used. Typical values for m/z tolerance are 0.015 m/z and 3 ppm.ADAP peak deconvolution:Peak list methods → peak detection → chromatogram deconvolution as shown in Next, the peaks from all chromatograms will be detected. Deconvolution is needed to separate the previously constructed chromatograms that span the entire duration. To run ADAP peak deconvolution select the chromatogram files on the left side of the screen and click deconvoluted as the suffix. From the drop-down box select Wavelets (ADAP). The methodology used for this project follows the Wavelet ADAP process which is the most recent and comprehensive method developed This will open a window with parameters for the deconvolution step as shown in After selecting Wavelets from the dropdown box, click on the ellipses to the right. This will open the algorithm parameter window shown in Below are descriptions of each algorithm parameter shown in S/N threshold:Signal to noise ratio threshold value greater than or equal to 7 will detect a small number of false peaks, this value compares peaks as 10x that of noise. 10 is a typical value to use for the Signal to noise threshold but can be adjusted based on results desired.S/N estimator:1.The intensity window SN, which is tested on LC-MS datasets, utilizes peak height as signal level and standard deviation of intensities around the peak as the noise level;This is the Signal to Noise estimator used for this project because we are working with LC-MS data and not GC-MS.2.Wavelet Coeff. SN, which is tested on the GC-MS datasets, utilizes continuous wavelet transform coefficients in order to estimate the signal and noise ratio. There is an analogous approach implemented in the R-package wmtsa User can choose one of two estimators of the signal-to-noise ratioMin feature height:Minimum height of a feature. Should be the same, or similar to, the value - min start intensity- set in the chromatogram building. This value should be set to match the noise level and group intensity threshold used throughout running the program. Again, this value will change depending on the noise determined for a certain data set.Coefficient/Area threshold:This is a threshold for the maximum coefficient (inner product) divided by the area under the curve of the feature. Filters out bad peaks. This number must be chosen by looking at examples using the show preview button at the bottom of the window. This is the best coefficient found by taking the inner product of the wavelet at the best scale and the peak, and then dividing by the area under the peak. This value should also be similar to the RT Wavelet range set below. Somewhere around 1 or 2 should work for most data sets but can range up to 100.Peak duration range:Peak duration range is the range of acceptable peak lengths. Analyze chromatogram to determine how long peaks last. Auto entered at 0–10 adjusted down to 0.0–1.0 after analyzing the chromatogram, this will vary from dataset to dataset. The smaller the range the more confidence in the identified peaks.RT wavelet range:Upper and lower bounds of retention times to be used for setting the wavelet scales. Choose a range that is similar to the range of peak widths expected to be found from the data. Auto entered at 0-0.1, the auto entered range typically works well but can be increased to 1 or 2 when the data calls for it.OK to return to the deconvolution parameter page. Then finish inputting those parameters shown below.Once all algorithm parameters are set, click m/z center calculation:Select “AUTO”m/z range for MS2 scan pairing (Da):Not entered/neededRT range for MS2 scan pairing (min):Not entered/neededPeak lists window on the right side of the screen. After the peak deconvolution program has run, a list of the chromatographic peaks will appear below the list of chromatograms in the Export data as csv fileFeature List Methods → Export/Import→ Export .csvThe data is now at the point where it can be exported to a .csv file so that it may be used in MFAssign R for formula assignment, identification, and interpretation. To export a csv, select the desired data and click This will display a parameter setting screen shown in Feature lists: this is the same in all parameter windows, make sure the desired set of peaks/data is selected.Filename: This is where the .csv will be stored; it is recommended to attach it to a blank excel sheet so it can be viewed and manipulated. It will still be stored as a .csv file. Make sure it is saved in a hard drive folder if using it in MFAssignRField separator: It will auto enter a comma in this field, change if needed.Export common elements: The MFAssignR code requires row m/z, and optionally, row retention time data , select these items under the common elements section.Export data file elements: The MFAssignR code also requires the Peak area data, select this item under the data file elements section.Export quantitation results and other information: Unnecessary, leave this box unchecked.Filter rows: Select “ALL”Preparing data for MFAssignR:not .ods or .xlsx and save the data on a hard drive in a folder that has a defined pathway, this will be needed for the MFAssignR code.Once the data from MZMine is successfully outputted into a .csv file,view the data in an Libre Office Calc The next section utilizes MFAssignR Install packages:install.packages(“dplyr”)install.packages(“tidyr”)install.packages(“colorRamps”)install.packages(“devtools”)install.packages(“ggplot2”)install.packages(“rmarkdown”)Installing packages is necessary to run the program. This only needs to be done once, can be commented out after they have been installed once. Lines can be “commented out” by inserting “#” in front of the desired line of code. A list of the packages needed for this program are listed below.setwd (“C:/Users/name/Drive/MFAssignR-master”)The next step is to write the call to install the MFAssignR program.devtools::install(“MFAssignR”)The next step is to set the working directory, this will be the folder that the MFAssignR package was saved to on the user's computer. Will be different to what is shown below.Data Loading:library(MFAssignR)library(ggplot2)library(dplyr)library(tidyr)Now loading the different necessary packages to do the formula assignment.setwd("C:/Users/lucyt/OneDrive/MFA Data")Data <- read.csvNow set the working directory to where the .csv file produced from MZmine was stored and load the datafile. Must attach the .csv file produced from MZmine to the word “data” to be called on throughout the rest of the code.SNRatio <- 0print("SNRatio")0Next step is to set the signal to noise ratio, this value allows the user to change the signal-to-noise ratio that will be multiplied by the estimated noise to determine the noise removal threshold. The SNRatio can be set from 0-10, but will likely be around 3. For the toxicity analysis project, the SNRatio was set to zero to ensure no legitimate peaks are mistaken for noise, since a majority of the noise was removed in the MZmine steps.Signal to noise assessment:Noise <-KMDNoiseThis is the signal to noise assessment section of the R markdown, demonstrating how to use the function KMDNoiseNoise [ ["KMD"]] #Plot showing the signal to noise estimation plotKMDN <-Noise [ ["Noise"]] #Saving the estimated noise as a global variable in the environmentKMDN #Printing the noise so that it can be seen in the final report.SNRatio = 0SNplotThis code shows how to extract the results of the KMDNoise function so that they can be usedThis will produce 3 output screens that are attached below. After analyzing The next figure window that will be produced from the Signal to Noise Assessment step is the KMDN value. This is a numerical representation of the estimated noise. This value will be saved in the global environment and used for future steps of the code.The last figure window that will be produced from this step is the noise index. This figure will change depending on what the SN Ratio is set to. If the SNRatio is set to 0, the visual will look like the one displayed in However, if the SNRatio is set to 1–10, the noise index will look more like the visual in Isotope identification:Isotopes <- IsoFiltRMono <- Isotopes [ ["Mono"]]Iso <- Isotopes [ ["Iso"]]This section shows the usage of the IsoFiltR function, which separates the single raw mass list into a list of likely monoisotopic masses ("Mono") and likely poly isotopic masses ("Iso"). The first line of code sets “Isotopes” as the results of the IsoFiltR function. Parameters for this include “Sulferr” and “Carberr”. This is the allowed error level when filtering sulfur and carbon. Typically, 3 is a good value for this but can be changed based on the mass accuracy of the instrument being used. The following two lines simply extract the two resultant mass lists from IsoFiltR and label them “Mono'' and “Iso” so that they can be used in later steps. There are no visual outputs for this section but extracted mass lists can be viewed in the upper right window/ the global environment as shown in Prelim assignment:Assign <- MFAssignCHO_RMDUnambig1 <- Assign [ ["Unambig"]] #Unambiguous molecular formula assignmentsAmbig1 <- Assign [ ["Ambig"]] #Ambiguous molecular formula assignmentsUnassigned <- Assign [ ["None"]] #Unassigned massesPlot1 <- Assign [ ["MSAssign"]] #Mass spectrum showing which peaks are assigned andunassigned in the spectrumPlot2 <- Assign [ ["Error"]] #Plot showing the error trend relative to mass forassignmentsPlot3 <- Assign [ ["MSgroups"]] #Mass spectrum showing the assigned molecular formulasPlot4 <- Assign [ ["VK"]] #O/C vs H/C plot showing the assigned molecular formulasPlot1Plot2Plot3Plot4Next is the preliminary assignment step, this will assign molecular formulas to the initial peaks before any recalibration is performed. Be sure this section of code is updated based on what the target assignments are. The following line shows how to use the CHO only version of formula assignment. It is typically done to find molecular formula series to be used in recalibration. The parameters given in the code are standard for ESI negative mode. rm(Plot1)rm(Plot2)rm(Plot3)rm(Plot4)rm(Unassigned)rm(Ambig1)gcThe next lines of code are included to clear up some of the memory to keep the markdown running as fast as possible.The first visual produced by the preliminary assignment code is the assignment mass spectrum, an example is shown in The next visual produced from this step of preliminary assignment is of the error plot, an example is shown in The following two visuals shown in Recalibration```{r, echo = FALSE, message = FALSE, warning = FALSE}Unambig1 <- Unambig1 Unambig1<- Unambig1%>%distinctcheck <- RecalList(Unambig1)The recalibration step requires the most user input. If available, representative samples should be run first to determine the set function “recalibrants” with confidence. The first two lines of code before the “check” step are optional but are useful for setting up recalibration in LC-MS runs because they remove duplicate mass/formula combinations (from isomers) that cause issues in recalibration calculations. Typically, these lines are set up to retain the most abundant example of each mass/formula combination. The goal of this is to remove all series that may have a Series Score less than 1, because a Series Score less than one causes issues for recalibration. This filtering step only impacts the selection of recalibrant series and the calculation of recalibration correction terms, all masses will be recalibrated in the function. This makes it easier to select formulas that cover the entire mass range spectrum.After running the above section of code, the user must run a qualitative check of “recalibrant” series and mass recalibration. This is done by examining the “check” page that can be found in the upper right-hand global workspace, also found in This is the data that will be analyzed to determine the series used to create the recalibrated mass spectrum. The overall goal is to get the longest continuous spectrum of ion masses covered by the series selected. This can be visually checked in Selecting series is determined on a set of parameters found in the column section of the “check” table.Examine the “check” table, in the environment to select 5 or 6 different chemicals to use in full assignment step. When selecting formulas, want mass ranges that cover a large range of the total mass spectrum (need to know highest and lowest mass range) while also examining the following parameters outlined in Test <- RecalPlot <- Test [ ["Plot"]]Plot #This plot can take time to generateMono2 <- Test [ ["Mono"]] #Recalibrated monoisotopic mass listIso2 <- Test [ ["Iso"]] #Recalibrated isotopic mass listsList <- Test [ ["RecalList"]] #The list of formulas/masses used as recalibrantsprint("mzRange")15When formulas are selected, insert them in the following code of the Recal function. Depending on confidence of formulas selected and peak and series scores, the mzRange is typically around 15 but can be increased if selected series are not completing the necessary range coverage. After the following lines of code are run the plot in Full assignmentIt is important that this section of code is updated with all necessary elements depending on assignment goals.Assign <- MFAssign_RMDUse of MFAssign_RMD for molecular formula assignment. The notable differences between the full and preliminary assignment steps are the option to add chemicals  to the assignment process, and the isotope error (set to 3), and having Sulfur check and MSMS set to “on”. Following the code below, besides what was indicated in It is important to note that the addition of more elements to formula assignment beyond C, H, and O can lead to increased ambiguity in the formula assignments due to more chemically feasible combinations of elements being possible, particularly at high masses Once as many formulas as possible are assigned and verified, the data can be run through once more with the addition of phosphorus. Phosphorus is difficult to assign because there are no isotopes, so it can swing the formula assignment results. When analyzing phosphorus assignments some unreasonable groups such as PNO2 will be present, these can be excluded from the confirmed formula assignment. This is because of the lack of enough oxygen to support what should likely be a phosphate group given the presence of phosphorus. This “staging” method may not be necessary for all data sets, some will produce satisfactory results with all chemicals included in the first run. Overall, when the output formula assignments are reviewed if any seem unreasonable they can be confirmed by comparing the mass in the data with the theoretical “exact mass” of that compound with an error range of around 3 ppm. The key is to always be critical of formula assignments from a chemical feasibility angle based on knowledge of the sample and analytical method, particularly when useres are first getting started.Unambig2 <- Assign [ ["Unambig"]]Ambig2 <- Assign [ ["Ambig"]]Unassigned2 <- Assign [ ["None"]]Plot1 <- Assign [ ["MSAssign"]]Plot2 <- Assign [ ["Error"]]Plot3 <- Assign [ ["MSgroups"]]Plot4 <- Assign [ ["VK"]]Plot1Plot2Plot3Plot4Extraction of data from the MFAssign_RMD function. It has the same format as the MFAssign CHO_RMD function. The code for outputting the final visuals and molecular formula assignments is below:Plots 1 through 4 match the ones produced in the preliminary assignment step shown in write.csvFinal step is to write the output csv code. This can be done by the following line of code for the desired data. Typically, the final results will be stored under the name “Unambig2” if the results call for it or if the user would like to analyze the ambiguous assignments as well, “Ambig2” can also be outputted as long as “Ambig=On” is specified in the MFAssign_RMD section of code. These object names can be changed by the user as desired.ToxAssign ToxAssign is broken into five modules, one module that automates execution and the other four that perform specific operations. The automation tool creates a directory for each sample, calls the PubChem module, toxic filter module, match module, the merge module three times, then the toxic filter module again. The PubChem module first matches potential toxic formulas from the list of data from MFAssignR against the local toxic table, then retrieves the full records from PubChem, then sorts them based on chosen records and writes them to files. The toxic filter module takes the toxic records and filters the records by acute toxicity and writes them to file. The match module matches any records not found on PubChem against a local table of records and writes the found records and still unfound records to file. Finally, the merge module takes the files from each sample and merges them together then writes them to file.To install this tool, the user will need to go to the Github page for this project PubChem:As is demonstrated in Then, in the same way as Next the module sorts the records by different fields that may or may not show up. To do so, the module loops through the fields of the record looking for either ones relating to toxicity or to food safety, checking in such a way as is shown in Toxic filter:The next module to be ran by the automation, as shown in Match:Next, the match module matches all records that were not found in the PubChem database against a local table of previously unfound records. Any that are matched are outputted to a file along with their safety class and any that are not are put back into the unfound file to be added to the local table .Fig. 33VMerge:The final module is the merge module, which takes the files from individual samples and merges them together for help in decisions about further analysis of the set, as well as easier analysis for repeated analysis of the same sample. This is done first as shown in Next, the module combines the negative data into the same object and finally moves out of the directory to begin again. As demonstrated in At the end the data object is written to an output file with the shared name of the in file as its name. This operation is performed three times: once for the toxic data, once for the unfound data, and once for the unchecked data. These are all merged to help choices for which standards to use to sample the entire data set, to help with adding to the remove data file, and to help with checking records that did not have any fields in the PubChem database.Toxic filter:As is shown in A goal of this paper was to replicate a procedure performed previously using an expensive and hard to obtain piece of software, Compound Discoverer, in an open source and easy to access and use format. This is to make the process not only less expensive but also more automated and easier to use, and thus, more accessible to under-resourced labs and more attractive to researchers to build upon this method in the future. To that end a part of this paper will be devoted to comparing this analysis to the previous analysis done using Compound Discoverer.The findings of this paper in regard to the volume of toxic compounds, while larger, are also still lacking one step of validation in their analysis. Also, it should be pointed out to this end this process finds only toxic compounds in the sample, while Compound Discoverer works to find all compounds within the sample, toxic or not. This process also uses different data than Compound Discoverer and different validation methods, so comparison becomes difficult. That said, looking at the comparison between the total ion intensity of the new data and the relative abundance of the old data shows a similar set of data .Fig. 36TLooking at This method still needs further validation and automation to overcome the gaps in the PubChem database as well as differences in naming conventions between the European Food Safety Authority ’s database and the NIH's database. For now, this is resolved with a manual search of records that do not match and a local storage of those records in a csv. Future extensions to this free software could work to use other services and online databases to match the European Food Safety Authority ’s recorded name to the NIH's recorded name and store the record in local JSON storage. This could also be used to store information about the compounds that have already been searched, allowing for faster processing and less strain on PubChem. Another potential upgrade includes building in functionality to analyze fracturing runs and retention time of compounds against standards using the MZCloud database The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Optimizing global connectivity in spatial networks, either through rewiring or adding edges, can increase the flow of information and increase the resilience of the network to failures. Yet, rewiring is not feasible for systems with fixed edges and optimizing global connectivity may not result in optimal local connectivity in systems where that is wanted. We describe the local network connectivity optimization problem, where costly edges are added to a systems with an established and fixed edge network to increase connectivity to a specific location, such as in transportation and telecommunication systems. Solutions to this problem maximize the number of nodes within a given distance to a focal node in the network while they minimize the number and length of additional connections. We compare several heuristics applied to random networks, including two novel planar random networks that are useful for spatial network simulation research, a real-world transportation case study, and a set of real-world social network data. Across network types, significant variation between nodal characteristics and the optimal connections was observed. The characteristics along with the computational costs of the search for optimal solutions highlights the need of prescribing effective heuristics. We offer a novel formulation of the genetic algorithm, which outperforms existing techniques. We describe how this heuristic can be applied to other combinatorial and dynamic problems. Spatial networks have become more popular as the interest in networks has spread into more fields and spatial data, and the computational power and methods to analyze it, have become more accessible. In terms of analysis, spatial network optimization has been at the forefront and focused on increasing network connectivity and information flow . HeuristWhile this optimization of spatial networks’ global characteristics, optimizing local existing network connectivity around a specific node or location with the introduction of costly new edges has not been explored and yet is important in several domains. For example, increasing an existing network’s connectivity around a focal node while minimizing the costs associated with the number and lengths of additional connections is essential in network layout planning for telecommunications and computer systems  and incrThe search for new edges that maximize connectivity to a focal node and minimize the costs of these new edges is not well understood and this search for optimal solutions can become costly when networks are large and complex. To fill this knowledge gap, we compare a set of heuristics to optimize local network connectivity applied to real-world networks and randomly generated ones. These heuristics are drawn from These optimization heuristics are then applied to randomly generated networks that vary in complexity and size to evaluate their efficacy in finding the optimal new connections that maximize local connectivity. Included in this set of random graphs we provide two novel formulations of random planar networks based on the Voronoi diagram and the Delaunay triangulation. To complement the random network analysis, the network connectivity optimization methods are also applied to two real-world case studies, one from urban transportation planning and another from social network analysis. In this study, we show that optimization heuristics are preferred for the analysis and practice due to the nonlinearity of the solution space and the optimal solution’s dependence on nodal characteristics, such as distance to the focal node. The novel genetic algorithm outperformed the other heuristics as it was able to move from suboptimal solutions and explore distance solutions quicker. This is important as researchers and engineers are working with networks or growing complexity and size.The organization of this paper is as follows. The next section describes the formulation of the connectivity problem in more detail, the local search methodology, and the optimization heuristics . The nodes that are within this distance are assigned to the ‘close’ set, NC ⊂ N, i.e., ν ∈ NC if d ≤ D and F ∈ NC  > D. When a new connection is added to the network, the shortest path distance from each distant node to the focal node is recalculated. If there are any distant nodes that are now within the threshold distance to the focal node they are assigned to the new set i and close node j, then k ∈ ND and d + d + d ≤ D  and the cost associated with the new connection is C. The optimal solution is the solution with the greatest benefit, or number of new nodes now within the distance to the focal node which can be expressed as the bi-objective function B = B and C < C, then the optimal additional edge is between i and j. For nondominated solutions, we select the solution that minimizes the distance to the so-called ideal point. The ideal point represents the solution that simultaneously maximizes the benefit and minimizes the cost. For the analysis in this paper the formulation of the objective function is as follows. For a new edge between i and j the number of nodes in C = d. Therefore, t) and terminate when the solutions converge, Ot = Ot−1, or the solution does not improve, Ot < Ot−1.For the description of the optimization methodology the following nomenclature will be used see . In conn∈ NC see . The nod, F) ≤ D . The benN the number of new connections to evaluate has an upper bound of N2∕4, therefore heuristics may be employed to identify (nearly) optimal solutions quicker than an exhaustive search as networks get larger. These optimization algorithms require a search space to explore and using nodal characteristics we create such a multidimensional solution space . The nodes that were directly connected to the node under evaluation were also used.For a network of size pace see . These npace see ; betweenpace see ; eigenvepace see ; and pagpace see . The weipace see . The weiThe following techniques were implemented for the network connectivity optimization study from their extensive use in optimization: hill climbing with random restart ; stochasExhaustive search (ES). The exhaustive search calculates the solution for every pair of distant and close nodes for a network (see Algorithm 1 in the Hill climbing (HC). The solution space was observed to be hilly from the exhaustive search results, so several modifications were introduced to the hill climbing technique to avoid getting stuck in suboptimal solutions (Algorithm 2 in the (HCS), an advanced search method based on HC, routine is also explored where the selection of nodes for the next iteration is randomly picked with (HCVN) where the size of the neighborhood starts with the nearest neighbors (η = 1) and is updated as follows: nmax is reached . As a meta-heuristic approach, the simulated annealing method randomly selects an initial solution from the solution space to avoid entrapment in a local optima. At each iteration, the heuristic evaluates the neighboring solutions and if it does not find an improved solution, it moves to a new solution with the following probability: Genetic algorithm (GA). The genetic algorithm begins with a population of P randomly selected solutions with a set of chromosomes composed of genes which represent the weights of selecting a neighbor and are all initialized to unity  while conducting a local neighborhood search.rossover . MutatioTo test the efficacy of these optimization heuristics in finding the optimal new network connections they were applied to randomly generated networks that vary in complexity and size. Several types of random graph networks were generated to analyze the efficacy of the optimization heuristics for systems with different topologies which are generally representative of naturally occurring and built systems: (1) Erdös-Rényi networks, (2) Watts-Strogatz networks, (3) Barabási and Albert networks, (4) Klemm and Eguílez networks, (5) Delaunay triangulation networks, and (6) Voronoi diagrams. Erdös-Rényi random networks are constructed by randomly creating connection between pairs of nodes with a probability . These npR is the removal probability and weighted by the normalized edge distance from the focal node. When edges are randomly removed from the connected Delaunay network or Voronoi network, with weights given by node distance from a focal node, these networks display some of the properties similarly found in the networks mentioned above, such as complexity and randomness. Yet, these networks have the added component of being planar and having edge weights that can be framed as physical distances.We also introduce two novel types of random planar network versions of the Voronoi diagram and the Delaunay triangulation  and S.1FTo compare the efficacy of different optimization methods for different network topologies, identifying the best set of parameters are critical. Parameter values were selected for each type of random network to ensure network complexity  and members who were within 0.5-miles of the event should be notified. The members within Grand Central Station are automatically notified of the event whereas those outside can be notified through their online social network if a member in their network is aware. The event is considered to be serious enough that those members aware of it will share the information on the social network. New connections are evaluated based on their number of additional nearby members who become aware of the event.Several finding are worthy to note regarding the performance of heuristic algorithms used in the analysis. First, there were consistent nonlinear relationships between the nodal characteristics and the quality of the solutions for each type of random network and the school networks see . There wThe results of the termination times and the optimal solutions deviations from the optimization heuristics applied to the random networks are summarized in The local network connectivity problem introduced in this study is relevant to a wide range of applications and is nontrivial as the number of potential solutions can become large even for small systems. This class of combinatorial optimization problem highlights the difficulty in determining local search routines a priori. When the exhaustive search routine was applied to random networks and the real-world networks, the optimal solutions were found to be related to nodal characteristics, which entails a great complexity to find optimal solutions. Therefore, the heuristics employed to reduce the computational costs utilized nodal characteristics to search for solutions. Yet, these nodal characteristics were nonlinearly related to the solutions. Given the example networks, it should be noted that distance to the focal node was consistently related to the quality of the solution as this lowers connectivity length costs, while centrality was intermittently correlated with solution quality it provides greater benefit through more connections. Aside from the distance characteristic, these nodal characteristics also varied in their correlation (sign and magnitude) with solution quality for different types of networks. This makes it difficult to exclude or prioritize specific nodal characteristics for local network connectivity optimization heuristics. This could arise from the four following issues: (a) the curse of dimensionality, i.e., large sparse subspaces in the solution space; (b) the nodal characteristics are highly correlated with each other; (c) outliers; and (d) the nodal characteristics are heterogeneous across the network. Results from the street networks in the transportation case-study found that the clustering coefficient was a poor measure due to the lack of triplets in the networks.The optimization heuristics save computational time but vary considerably in their ability to find (near) optimal solutions. The stochastic hill climbing search was not effective due to the large neighborhood search space explored. In our experiment, the number of solutions checked at each iteration is >300 and resulted in a skewed probability distribution of objective values favoring the selection of low values. This degraded the efficiency of the method resulting in the selection of poor solutions. The variable neighborhood search method was similarly not reliable because of the significantly large neighborhood search space , and had intermediate results with cost and benefit deviations. The simulated annealing heuristic consistently took longer to converge than the other optimization methods from the exploration of suboptimal solutions prior to moving towards better solutions, yet it was able to converge to values close to the optimal solution.The computational costs and the variance in the importance of nodal characteristics for the random networks and real-world systems highlights the need for a heuristic that is able to quickly and effectively explore the solution space without getting stuck in a local optimum. The genetic algorithm provided in this work offers a solution to these issues and outperformed the other algorithms in terms of the consistently higher solution precision and accuracy. The genetic algorithm is able to dynamically reduce the size of the neighborhood search space and what variables to analyze. This reduction in the local solution search space allows the genetic algorithm presented here to converge on solutions near the optimal in a timely fashion. The heuristic is also able to compare solutions from distant search spaces with nonzero probability, thereby avoiding local optimums. The experiments indicate the power of biologically inspired algorithms to effectively explore multidimensional spaces  and their potential use in a wide variety of disciplines, including the specific applications for planning and crisis management presented above. The combinatorial optimization techniques employed here to identify and evaluate new street connections can also complement the optimization approaches used for other transportation planning problems, such as greenway planning , bus stoThere are several research directions from these proposed methods. Application of these methods and heuristics can be tested on multi-level networks, such as telecommunication systems, higher dimensional real-world networks (transportation networks with elevation), directed networks, and additional planar random networks  and this should be conducted. Different distance measures to the focal node, such as the Hamming distance, could also be evaluated for different applications, and other real world examples should be used for analysis. The methods presented here simplified the costs and did not account for many real-world barriers that may restrict optimal new connections found through the heuristics. For example, the transportation case study did not include legal considerations, such as right-of-way, or physical barriers, such as highways or rivers. Furthermore, the methods presented here do not evaluate whether the new connections intersect existing edges, as in existing transportation networks, and attempts to incorporate such a feature resulted in unrealistic computational times.10.7717/peerj-cs.605/supp-1Supplemental Information 1Click here for additional data file.10.7717/peerj-cs.605/supp-2Supplemental Information 2Matlab code for each algorithm shown in the Supplemental Materials.Click here for additional data file.10.7717/peerj-cs.605/supp-3Supplemental Information 3Matlab files containing the adjacency matrices for the ten schools used in the real world case study.Click here for additional data file.


Based on laser Thomson scattering (TS) measurements and finite element method (FEM) simulations of electron density in inductively coupled plasma (ICP), the simulated local pressure calibration curves of ICP generator are obtained by comparing the experimental and simulated electron density distributions and maxima. The equation coefficients of theoretical model associated with the ICP generator experimental system can be obtained by fitting the simulation curve with the least square method, and the theoretical pressure calibration curves under different absorbed powers can be further obtained. Combined with the vacuum gauge measurements, both the simulated and theoretical pressure calibration curves can give the true local pressure in the plasma. The results of the local pressure calibration at the different absorbed powers show that the density gradient from the vacuum gauge sensor to the center of the coil in ICP generator cavity becomes larger with the increase of electron density, resulting in a larger gap between the measured value and the pressure calibration value. This calibration method helps to grasp the local pressure of ICP as an external control factor and helps to study the physicochemical mechanism of ICP in order to achieve higher performance in ICP etching, material modification, etc. However, accurate and skilled use of these processes requires a clear understanding of the various ICP control factors such as ICP internal pressure and absorbed power. These control factors can sensitively affect the electron density, electron temperature and other physical parameters of ICP, thus profoundly affect the accuracy of production process. Especially important, the local pressure in a low-pressure plasma affects the mean free path of collisions between electrons and neutral particles, which is closely related to local and non-local dynamic characteristics6, electron heating and heating efficiency, electron energy distribution7 and electron density8, discharge mode (E or H mode)10 and so on. Therefore, the local pressure is a very important external parameter of low-pressure plasma, especially in terms of accuracy control during the plasma process. As an electrical parameter, absorbed power can be accurately measured by tracing its source. However, the local pressure in ICP is difficult to be accurately measured because of the distance between the sensor of the vacuum gauge and the plasma region to be measured. To understand the actual pressure in the plasma, it is necessary to calibrate the local pressure in ICP. This provides quantitative pressure parameters for studying the physicochemical mechanisms of various processes in ICP.Inductively Coupled Plasma (ICP) has been widely used for crystal etching12 and there is a close correlation between electron density and electron temperature13. So we can focus on comparing the distribution of electron density to determine whether the external conditions of the simulations and experiments in this paper are consistent. However, this comparison depends on accurate and reliable measurements of electron density. At present, the commonly used measurement methods for ICP electron density include Langmuir probe method14, emission spectroscopy15, microwave interferometry16, laser interferometry17 and laser Thomson scattering (TS) measurement18. Among these techniques for measuring plasma electron density, the TS can measure the electron density in a given region without disturbing the plasma, and has very high spatial and temporal resolution19. It is recognized as one of the most accurate electron density measurements available. So in this paper we use TS to measure the electron density distribution of ICP.Electron density and electron temperature are the main physical parameters to characterize the plasma, so the high agreement between their finite element simulation results with the experimental results can fully guarantee the reliability of the computational model. Both simulation and experiment reveal that the gradient in the electron density distribution of ICP is very pronounced under H-mode discharge condition. The plasma is firmly bound inside of the coil by the inductor coil. As a statistical physical quantity, the measured or simulated electron temperature has no significant gradient distribution. Obviously, the identification of electron density is much higher than that of electron temperature20 measured the spatial distribution of neutral gas temperature and total pressure, electron pressure and neutral pressure by inserting a thin tube probe with diameter of 3.15 mm into the plasma. However, this interventional measurement itself interferes with the local environment of the measurement area, making it difficult to achieve accurate measurements. In order to avoid the measurement uncertainty caused by interventional measurements, a new method of ICP local pressure calibration combining TS experiments and finite element method (FEM) simulations is proposed in this paper. First, the electron density distribution of ICP is measured non-invasively by laser TS. The ICP pressure to be calibrated is measured by a vacuum gauge probe  at the gas inlet of the ICP generator. The absorbed power of the ICP is the product of the input power of a 13.56 MHz RF power supply and the power transfer efficiency. As the power and pressure change, the plasma density changes, and the plasma load on the matching network varies accordingly. To compare the experimental and simulated values of electron density distribution at a specific absorbed power, the capacitance in the matching circuit is experimentally adjusted by Smith chart to match the internal impedance of the RF power supply so that a given power such as 400 W, 350 W and 300 W is transmitted into the plasma. The power transfer efficiency is 96.3%, 94.9% and 93.0% at an absorbed power of 400 W, 350 W and 300 W, respectively, as measured by a similar method in reference21. Second, a FEM model with the same physical dimensions as the ICP generator experimental setup is established. At the same absorbed power as the experiment, the pressure measured by the vacuum gauge is taken as the reference starting point, and the pressure is scanned in the lower pressure direction. The resulting simulation yields a series of spatial distributions of the electron density with pressure in the ICP generator cavity. Third, this series of plasma electron density simulation data is used as the true value of the measurement and compared with the electron density data measured by TS. When both the experimental and simulated electron density distributions and maxima are within the error tolerance, the pressure value in the simulation can be determined as the corrected value for the pressure measurement. This completes a single pressure data calibration of the ICP generator. This cycle can be repeated to complete a pressure calibration simulation curve for a specific ionized gas at a given absorbed power. At last, combined with the theoretical model, the equation coefficients associated with the ICP generator experimental system are obtained by fitting the pressure calibration simulation curve using the least square method, so as to obtain the theoretical pressure calibration curves for the different absorbed powers. Both simulated and theoretical calibration curves can be used for the local pressure calibration in low-pressure plasma processes.Accurate measurement and calibration of pressure, especially local pressure, has rarely been reported in previous research work. In 2007, Shimada22. When a laser with a wavelength of λ0 enters plasma, the free electrons in the plasma radiate electromagnetic waves under the action of the incident laser electric field. Since TS is elastic scattering, the wavelength of the scattered light is the same as that of the incident laser. However, the Doppler Effect is obvious due to the fast electron motion, so the Doppler broadening of TS spectrum appears. The electron density and electron temperature of plasma can be obtained by measuring TS spectrum and data post-processing.Laser TS technique can measure the electron density and electron temperature of plasma by measuring the secondary radiation emitted by the interaction between free electrons and the incident laserAs shown in Fig. 18 m-3 ~ 1019 m-3.In order to keep the flexibility of space, we use optical fiber to collect the TS signal and transmit them to the slit of TGS system in this scheme. The collection direction of optical fiber is at a 90-degree angle to the laser transmission direction and the axial direction of the plasma beam. A set of plano-convex lenses is arranged in front of the fiber to focus the laser-plasma interaction region to the fiber entrance. To avoid Rayleigh scattering from the air and reflections from other surfaces, the collection end of the optical fiber and the collection lens are placed in a dark chamber. The TS signal is transmitted to the TGS system through the optical fiber, passing through the grating, lens and other optical devices in the TGS system, and finally is imaged or formed into spectrum on the ICCD camera. The DG645 is a time controller that synchronizes the laser pulse with the ICCD camera shutter and optimizes the detected signal by changing the ICCD camera exposure time. The TGS system can effectively filter out Rayleigh scattering and stray light, which is beneficial to the extraction of laser TS signal. The electron density measured with TS in this experiment ranges from 10The electron density distribution of ICP measured with laser TS at 350 W absorbed power and 65 Pa pressure (measured by vacuum gauge) is shown below Fig. .Experimental parameters such as ICP pressure and absorbed power were recorded during the experiment. The pressure measured in the experiment was used as the reference starting value of the pressure parameter scanning in the plasma simulation. The parameters such as absorbed power, coil size, and cavity dimensions were kept consistent with the experimental system when the ICP was simulated.24 (V m s)−1.The Frequency-Transient study is used to simulate ICP with COMSOL software in this paper. The diffusion model is Mixture-averaged including migration in electric field. Reduced electron transport characteristics with local energy approximation are used in plasma properties settings. The electron energy distribution function is set as Maxwellian. And the model temperature is specified as 300 K. The electron transport property is restricted to specify mobility only, and the isotropic reduced electron mobility is set to 4 × 10The finite element model of ICP is developed according to the physical dimensions of its generator experimental system. Since the axisymmetric tubular structure is to be simulated, a 2D axisymmetric simulation model is established to save computational resources, as shown in Fig. 15 m−3, and the initial average electron energy is set as 5 V. The coil is set to a single wire model in the magnetic field settings, specifying the absorbed power. The model is filled with triangle meshes of 17,356 cells and the minimum cell is set to 45 µm. Boundary layers have been created next to the walls in order to solve properly the high gradients formed during the simulation. Each boundary layer is divided into 5 sub-layers and smoothed over to the interior mesh. The mesh has been refined several times. The external boundaries of the whole geometry are magnetically insulated excepting the symmetry axis.Convective flux is not considered and therefore no gas inflow and outflow are used. The air surrounding the coil is considered to possess vacuum characteristics. During the simulation, the electron neutral initial value of argon ion and the neutral mass of argon atom are constrained. The wall surrounding the plasma is grounded and the general wall reflection coefficient is set as 0.2. The initial electron density is set as 1 × 10–6 s and 10–1 s.When the ICP device is first energized, all the power dissipated is in the coil. After about 1 µs, the plasma ignition begins and as the neutral gas atoms split into electrons and ions, the electrons begin to absorb more and more power and further ionize the neutral gas. The local electron density in the center reaches instantaneous maximum value in more than 30 µs, and then reaches a stable state with the diffusion of electrons. In order to ensure that the ICP could reach a stable state, the calculation time should be set long enough. The electron density distribution evolution over time varies with pressure and absorbed power, but 0.1 s is sufficient to achieve a stable distribution in our study. In order to understand the evolution process of ionization over time and ensure that the electron density distribution reaches a stable state, we have performed simulations with 20 time interpolation points between 10The analysis of the plasma electron density distribution was carried out after convergence. As the plasma electron density is axisymmetric, cut line data should be collected along the axis of the glass tube and perpendicular to the axis of the glass tube at the center of the coil as shown in Fig. During the simulation, a series of electron density distribution data for a certain absorbed power can be obtained by scanning the pressure parameters. Capturing their maximum values and plotting them with a series of pressure-related dotted lines, a local pressure correction curve for the ICP center can be obtained, as shown in Fig. 23The dynamic behavior of the electron density is characterized by the ambipolar diffusion of electrons and ions. The plasma-electron rate equation based on the ambipolar diffusion is obtained from the continuity equationDA is the ambipolar diffusion coefficient and ne is the electron density. In cold plasmas like ICP, k is Planck's constant, Te is the electron temperature of plasma, M is the rest-mass of ions24. And Ti is the ion temperature. The right term Q in Eq. (Ki is the average ionization rate coefficient, including the ionization of the ground state and several excited states. 25. Electrons scatter isotropically in collisions with neutral particles, thermalizing the energy they gain. This process is repeated until their temperature Te is determined. Thus, the electron temperature is proportional to the product of the mean free path and the total electric field inside the discharge tube26 expressed asξ is the thermalization form factor of electron energy. Substituting Eq. (24where mentclass2pt{minimmentclass2pt{minim27We assume that ting Eq.  into Eq.A and C are directly proportional to the total electric field E. As shown in Table For the present experimental conditions, the neutral density Taking three absorbed powers of 300 W, 350 W and 400 W as examples, the relationships between the maximum electron densities measured by laser TS and the pressures measured by vacuum gauge have been compared with the simulated and theoretical results, as shown in Fig. p and the density n can be related by a thermodynamic equation T is the temperature of the particle27. We can assume that the neutral density on the vacuum gauge sensor The pressure difference between the experimental and calibrated values shown in Fig. In this paper, the plasma electron density distributions of ICP are obtained by means of TS experiment and FEM simulation. The simulated calibration value for the pressure measured by the vacuum gauge has been determined by comparing simulated and experimental results of the electron density distribution and the maximum electron density under the same conditions including the absorbed power. As a result, calibration simulation curve has been obtained for a given absorbed power. A more accurate theoretical equation for the relationship between maximum electron density and pressure has been derived for the first time. And the simulation curves have been matched with the theoretically derived equation to obtain the exact equation parameters for given powers. As calibration examples, three theoretical fitting curves have been used to calibrate the pressures at 300 W, 350 W and 400 W respectively for the same ICP generator. The pressure calibration results show that the pressure gradient from the vacuum gauge sensor to the center of the coil in ICP generator cavity become larger with the increase of electron density. The control of the local pressure at the ICP center point will help to improve the accuracy of low-pressure plasma processes such as plasma thin film deposition, etching and material surface treatment in the future.


The other authors have stated explicitly that there are no conflicts of interest in connection with this article.To the Editor,In their recently published study, Kuroda et al.Kuroda et al. did not show summary statistics of the duration of the follow‐up period of each case. Those periods are vital information for the analyses. For example, the statistical weight is more massive in a count of 10 EHVs out of 14 days than the same count out of 730 days. Cancer patients usually have a shorter follow‐up period because their prognoses are worse than noncancer patients. Kato et al.In conclusion, this study is informative because it demonstrated the factors of emergency home visits. However, we think they need to implement the duration of the follow‐up period for each case into their analyses to determine the appropriate statistical weight of each factor.


We wish to make the following correction to the published paper “Effects of Transient Loss of Vision on Head and Eye Movements during Visual Search in a Virtual Environment” .We have identified a flaw in the implementation of a latency mitigation strategy for our gaze-contingent protocol written in Unity3D. As a result, the maximum latency is now estimated to be 30 ms instead of 15 ms, which should not affect any of the results originally published but should be noted for further reference.