Model-Free Reconstruction of Excitatory Neuronal Connectivity from Calcium Imaging Signals

A systematic assessment of global neural network connectivity through direct electrophysiological assays has remained technically infeasible, even in simpler systems like dissociated neuronal cultures. We introduce an improved algorithmic approach based on Transfer Entropy to reconstruct structural connectivity from network activity monitored through calcium imaging. We focus in this study on the inference of excitatory synaptic links. Based on information theory, our method requires no prior assumptions on the statistics of neuronal firing and neuronal connections. The performance of our algorithm is benchmarked on surrogate time series of calcium fluorescence generated by the simulated dynamics of a network with known ground-truth topology. We find that the functional network topology revealed by Transfer Entropy depends qualitatively on the time-dependent dynamic state of the network (bursting or non-bursting). Thus by conditioning with respect to the global mean activity, we improve the performance of our method. This allows us to focus the analysis to specific dynamical regimes of the network in which the inferred functional connectivity is shaped by monosynaptic excitatory connections, rather than by collective synchrony. Our method can discriminate between actual causal influences between neurons and spurious non-causal correlations due to light scattering artifacts, which inherently affect the quality of fluorescence imaging. Compared to other reconstruction strategies such as cross-correlation or Granger Causality methods, our method based on improved Transfer Entropy is remarkably more accurate. In particular, it provides a good estimation of the excitatory network clustering coefficient, allowing for discrimination between weakly and strongly clustered topologies. Finally, we demonstrate the applicability of our method to analyses of real recordings of in vitro disinhibited cortical cultures where we suggest that excitatory connections are characterized by an elevated level of clustering compared to a random graph (although not extreme) and can be markedly non-local.

Effect of anionic salts in concentrate mixture and calcium intake on some blood and urine minerals, acid-base balance and feed intake of dry pregnant cows on grass silage based feeding

Selostus: Kationi-anionitasapaino ja kalsiumin saanti ummessaolevien lypsylehmien säilörehuruokinnassa

Intercellular calcium waves in primary cultured rat mesenteric smooth muscle cells are mediated by connexin43.

Intercellular Ca(2+) wave propagation between vascular smooth muscle cells (SMCs) is associated with the propagation of contraction along the vessel. Here, we characterize the involvement of gap junctions (GJs) in Ca(2+) wave propagation between SMCs at the cellular level. Gap junctional communication was assessed by the propagation of intercellular Ca(2+) waves and the transfer of Lucifer Yellow in A7r5 cells, primary rat mesenteric SMCs (pSMCs), and 6B5N cells, a clone of A7r5 cells expressing higher connexin43 (Cx43) to Cx40 ratio. Mechanical stimulation induced an intracellular Ca(2+) wave in pSMC and 6B5N cells that propagated to neighboring cells, whereas Ca(2+) waves in A7r5 cells failed to progress to neighboring cells. We demonstrate that Cx43 forms the functional GJs that are involved in mediating intercellular Ca(2+) waves and that co-expression of Cx40 with Cx43, depending on their expression ratio, may interfere with Cx43 GJ formation, thus altering junctional communication.

Effects of surface application of calcium-magnesium silicate and gypsum on soil fertility and sugarcane yield

Lime application recommendations for amendment of soil acidity in sugarcane were developed with a burnt cane harvesting system in mind. Sugarcane is now harvested in most areas without burning, and lime application for amendment of soil acidity in this system in which the sugarcane crop residue remains on the ground has been carried out without a scientific basis. The aim of this study was to evaluate the changes in soil acidity and stalk and sugar yield with different rates of surface application of calcium, magnesium silicate, and gypsum in ratoon cane. The experiment was performed after the 3rd harvest of the variety SP 81-3250 in a commercial green sugarcane plantation of the São Luiz Sugar Mill (47º 25' 33" W; 21º 59' 46" S), located in Pirassununga, São Paulo, in southeast Brazil. A factorial arrangement of four Ca-Mg silicate rates (0, 850, 1700, and 3400 kg ha-1) and two gypsum rates (0 and 1700 kg ha-1) was used in the experiment. After 12 months, the experiment was harvested and technological measurements of stalk and sugar yield were made. After harvest, soil samples were taken at the depths of 0.00-0.05, 0.05-0.10, 0.10-0.20, 0.20-0.40, and 0.40-0.60 m in all plots, and the following determinations were made: soil pH in CaCl2, organic matter, P, S, K, Ca, Mg, H+Al, Al, Si, and base saturation. The results show that the application of gypsum reduced the exchangeable Al3+ content and Al saturation below 0.05 m, and increased the Ca2+ concentration in the whole profile, the Mg2+ content below 0.10 m, K+ below 0.4 m, and base saturation below 0.20 m. This contributed to the effect of surface application of silicate on amendment of soil acidity reaching deeper layers. From the results of this study, it may be concluded that the silicate rate recommended may be too low, since the greater rates used in this experiment showed greater reduction in soil acidity, higher levels of nutrients at greater depths and an increase in stalk and sugar yield.

HR-243 Production and Evaluation of Calcium Magnesium Acetate, Febrauray 19, 2001

Calcium magnesium acetate (CMA) has been identified by Bjorksten Research Laboratories as an environmentally harmless alternative to sodium or calcium chloride for deicing highways. Their study found CMA to be noncorrosive to steel, aluminum and zinc with little or no anticipated environmental impact. When used, it degrades into elements found in abundance in nature. The deicing capabilities were found to be similar to sodium chloride. The neutralized CMA they produced did cause scaling of PC concrete, but they did not expect mildly alkaline CMA to have this effect. In the initial investigation of CMA at the Iowa DOT laboratory, it was found that CMA produced from hydrated lime and acetic acid was a light, fluffy material. It was recognized that a deicer in this form would be difficult to effectively distribute on highways without considerable wind loss. A process was developed to produce CMA in the presence of sand to increase particle weight. In this report the product of this process, which consists of sand particles coated with CMA, is referred to as "CMA deicer". The mixture of salts, calcium magnesium acetate, is referred to as "CMA". The major problems with CMA for deicing are: (1) it is not commercially available, (2) it is expensive with present production methods and (3) there is very little known about how it performs on highways under actual deicing conditions. In view of the potential benefits this material offers, it is highly desirable to find solutions or answers to these problems. This study provides information to advance that effort.

TORC1 is a calcium- and cAMP-sensitive coincidence detector involved in hippocampal long-term synaptic plasticity.

A key feature of memory processes is to link different input signals by association and to preserve this coupling at the level of synaptic connections. Late-phase long-term potentiation (L-LTP), a form of synaptic plasticity thought to encode long-term memory, requires gene transcription and protein synthesis. In this study, we report that a recently cloned coactivator of cAMP-response element-binding protein (CREB), called transducer of regulated CREB activity 1 (TORC1), contributes to this process by sensing the coincidence of calcium and cAMP signals in neurons and by converting it into a transcriptional response that leads to the synthesis of factors required for enhanced synaptic transmission. We provide evidence that TORC1 is involved in L-LTP maintenance at the Schaffer collateral-CA1 synapses in the hippocampus.

Régulation du canal à sodium épithélial par le sodim extracellulaire et développement d'un microsystème intégré pour l'étude électrophysiologique sur ovocytes de "Xenopus laevis"

Résumé : La première partie de ce travail de thèse est consacrée au canal à sodium épithélial (ENaC), l'élément clé du transport transépithélial de Na+ dans le néphron distal, le colon et les voies aériennes. Ce canal est impliqué dans certaines formes génétiques d'hypo- et d'hypertension (PHA I, syndrome de Liddle), mais aussi, indirectement, dans la mucoviscidose. La réabsorption transépithéliale de Na+ est principalement régulée par des hormones (aldostérone, vasopressine), mais aussi directement par le Na+, via deux phénomènes distincts, la « feedback inhibition » et la « self-inhibition » (SI). Ce second phénomène est dépendant de la concentration de Na+ extracellulaire, et montre une cinétique rapide (constante de temps d'environ 3 s). Son rôle physiologique serait d'assurer l'homogénéité de la réabsorption de Na+ et d'empêcher que celle-ci soit excessive lorsque les concentrations de Na+ sont élevées. Différents éléments appuient l'hypothèse de la présence d'un site de détection de la concentration du Na+ extracellulaire sur ENaC, gouvernant la SI. L'objectif de ce premier projet est de démontrer l'existence du site de détection impliqué dans la SI et de déterminer ses propriétés physiologiques et sa localisation. Nous avons montré que les caractéristiques de la SI (en termes de sélectivité et affinité ionique) sont différentes des propriétés de conduction du canal. Ainsi, nos résultats confirment l'hypothèse de l'existence d'un site de détection du Na+ (responsable de la transmission de l'information au mécanisme de contrôle de l'ouverture du canal), différent du site de conduction. Par ailleurs, ce site présente une affinité basse et indépendante du voltage pour le Na+ et le Li+ extracellulaires. Le site semble donc être localisé dans le domaine extracellulaire, plutôt que transmembranaire, de la protéine. L'étape suivante consiste alors à localiser précisément le site sur le canal. Des études précédentes, ainsi que des résultats préliminaires récemment obtenus, mettent en avant le rôle dans la self-inhibition du premiers tiers des boucles extracellulaires des sous-unités α et γ du canal. Le second projet tire son origine des limitations de la méthode classique pour l'étude des canaux ioniques, après expression dans les ovocytes de Xenopus laevis, par la méthode du voltage-clamp à deux électrodes, en particulier les limitations dues à la lenteur des échanges de solutions. En outre, cette méthode souffre de nombreux désavantages (manipulations délicates et peu rapides, grands volumes de solution requis). Plusieurs systèmes améliorés ont été élaborés, mais aucun ne corrige tous les désavantages de la méthode classique Ainsi, l'objectif ici est le développement d'un système, pour l'étude électrophysiologique sur ovocytes, présentant les caractéristiques suivantes : manipulation des cellules facilitée et réduite, volumes de solution de perfusion faibles et vitesse rapide d'échange de la perfusion. Un microsystème intégré sur une puce a été élaboré. Ces capacités de mesure ont été testées en utilisant des ovocytes exprimant ENaC. Des résultats similaires (courbes IV, courbes dose-réponse au benzamil) à ceux obtenus avec le système traditionnel ont été enregistrés avec le microsystème. Le temps d'échange de solution a été estimé à ~20 ms et des temps effectifs de changement ont été déterminés comme étant 8 fois plus court avec le nouveau système comparé au classique. Finalement, la SI a été étudiée et il apparaît que sa cinétique est 3 fois plus rapide que ce qui a été estimé précédemment avec le système traditionnel et son amplitude de 10 à 20 % plus importante. Le nouveau microsystème intégré apparaît donc comme adapté à la mesure électrophysiologique sur ovocytes de Xenopus, et possèdent des caractéristiques appropriées à l'étude de phénomènes à cinétique rapide, mais aussi à des applications de type « high throughput screening ». Summary : The first part of the thesis is related to the Epithelial Sodium Channel (ENaC), which is a key component of the transepithelial Na+ transport in the distal nephron, colon and airways. This channel is involved in hypo- and hypertensive syndrome (PHA I, Liddle syndrome), but also indirectly in cystic fibrosis. The transepithelial reabsorption of Na+ is mainly regulated by hormones (aldosterone, vasopressin), but also directly by Na+ itself, via two distinct phenomena, feedback inhibition and self-inhibition. This latter phenomenon is dependant on the extracellular Na+ concentration and has rapid kinetics (time constant of about 3 s). Its physiological role would be to prevent excessive Na+ reabsorption and ensure this reabsorption is homogenous. Several pieces of evidence enable to propose the hypothesis of an extracellular Na+ sensing site on ENaC, governing self-inhibition. The aim of this first project is to demonstrate the existence of the sensing site involved in self-inhibition and to determine its physiological properties and localization. We show self-inhibition characteristics (ionic selectivity and affinity) are different from the conducting properties of the channel. Our results support thus the hypothesis that the Na+ sensing site (responsible of the transmission of the information about the extracellular Na+ concentration to the channel gating mechanism), is different from the channel conduction site. Furthermore, the site has a low and voltage-insensitive affinity for extracellular Na+ or Li+. This site appears to be located in the extracellular domain rather than in the transmembrane part of the channel protein. The next step is then to precisely localize the site on the channel. Some previous studies and preliminary results we recently obtained highlight the role of the first third of the extracellular loop of the α and γ subunits of the channel in self-inhibition. The second project originates in the limitation of the classical two-electrode voltageclamp system classically used to study ion channels expressed in Xenopus /aevis oocytes, in particular limitations related to the slow solution exchange time. In addition, this technique undergoes several drawbacks (delicate manipulations, time consumption volumes). Several improved systems have been built up, but none corrected all these detriments. The aim of this second study is thus to develop a system for electrophysiological study on oocytes featuring an easy and reduced cell handling, small necessary perfusion volumes and fast fluidic exchange. This last feature establishes the link with the first project, as it should enable to improve the kinetics analysis of self-inhibition. A PDMS chip-based microsystem has been elaborated. Its electrophysiological measurement abilities have been tested using oocytes expressing ENaC. Similar measurements (IV curves of benzamil-sensitive currents, benzamil dose-response curves) have been obtained with this system, compared to the traditional one. The solution exchange time has been estimated at N20 ms and effective exchange times (on inward currents) have been determined as 8 times faster with the novel system compared to the classical one. Finally, self-inhibition has been studied and it appears its kinetics is 3 times faster and its amplitude 10 to 20 % higher than what has been previously estimated with the traditional system. The novel integrated microsystem appears therefore to be convenient for electrophysiological measurement on Xenopus oocytes, and displays features suitable for the study of fast kinetics phenomenon, but also high throughput screening applications. Résumé destiné large public : Le corps humain est composé d'organes, eux-mêmes constitués d'un très grand nombre de cellules. Chaque cellule possède une paroi appelée membrane cellulaire qui sépare l'intérieur de cette cellule (milieu intracellulaire) du liquide (milieu extracellulaire) dans lequel elle baigne. Le maintien de la composition stable de ce milieu extracellulaire est essentiel pour la survie des cellules et donc de l'organisme. Le sodium est un des composants majeurs du milieu extracellulaire, sa quantité dans celui-ci doit être particulièrement contrôlée. Le sodium joue en effet un rôle important : il conditionne le volume de ce liquide extracellulaire, donc, par la même, du sang. Ainsi, une grande quantité de sodium présente dans ce milieu va de paire avec une augmentation du volume sanguin, ce qui conduit l'organisme à souffrir d'hypertension. On se rend donc compte qu'il est très important de contrôler la quantité de sodium présente dans les différents liquides de l'organisme. Les apports de sodium dans l'organisme se font par l'alimentation, mais la quantité de sodium présente dans le liquide extracellulaire est contrôlée de manière très précise par le rein. Au niveau de cet organe, on appelle urine primaire le liquide résultant de la filtration du sang. Elle contient de nombreuses substances, des petites molécules, dont l'organisme a besoin (sodium, glucose...), qui sont ensuite récupérées dans l'organe. A la sortie du rein, l'urine finale ne contient plus que l'excédent de ces substances, ainsi que des déchets à éliminer. La récupération du sodium est plus ou moins importante, en fonction des ajustements à apporter à la quantité présente dans le liquide extracellulaire. Elle a lieu grâce à la présence de protéines, dans les membranes des cellules du rein, capables de le transporter et de le faire transiter de l'urine primaire vers le liquide extracellulaire, qui assurera ensuite sa distribution dans l'ensemble de l'organisme. Parmi ces protéines « transporteurs de sodium », nous nous intéressons à une protéine en particulier, appelée ENaC. Il a été montré qu'elle jouait un rôle important dans cette récupération de sodium, elle est en effet impliquée dans des maladies génétiques conduisant à l'hypo- ou à l'hypertension. De précédents travaux ont montré que lorsque le sodium est présent en faible quantité dans l'urine primaire, cette protéine permet d'en récupérer une très grande partie. A l'inverse, lorsque cette quantité de sodium dans l'urine primaire est importante, sa récupération par le biais d'ENaC est réduite. On parle alors d'autorégulation : la protéine elle-même est capable d'adapter son activité de transport en fonction des conditions. Ce phénomène d'autorégulation constitue a priori un mécanisme préventif visant à éviter une trop grande récupération de sodium, limitant ainsi les risques d'hypertension. La première partie de ce travail de thèse a ainsi consisté à clarifier le mécanisme d'autorégulation de la protéine ENaC. Ce phénomène se caractérise en particulier par sa grande vitesse, ce qui le rend difficile à étudier par les méthodes traditionnelles. Nous avons donc, dans une deuxième partie, développé un nouveau système permettant de mieux décrire et analyser cette « autorégulation » d'ENaC. Ce second projet a été mené en collaboration avec l'équipe de Martin Gijs de l'EPFL.

Congenital ataxia and hemiplegic migraine with cerebral edema associated with a novel gain of function mutation in the calcium channel CACNA1A.

Mutations in the CACNA1A gene, encoding the α1 subunit of the voltage-gated calcium channel CaV2.1 (P/Q-type), have been associated with three neurological phenotypes: familial and sporadic hemiplegic migraine type 1 (FHM1, SHM1), episodic ataxia type 2 (EA2), and spinocerebellar ataxia type 6 (SCA6). We report a child with congenital ataxia, abnormal eye movements and developmental delay who presented severe attacks of hemiplegic migraine triggered by minor head traumas and associated with hemispheric swelling and seizures. Progressive cerebellar atrophy was also observed. Remission of the attacks was obtained with acetazolamide. A de novo 3bp deletion was found in heterozygosity causing loss of a phenylalanine residue at position 1502, in one of the critical transmembrane domains of the protein contributing to the inner part of the pore. We characterized the electrophysiology of this mutant in a Xenopus oocyte in vitro system and showed that it causes gain of function of the channel. The mutant CaV2.1 activates at lower voltage threshold than the wild type. These findings provide further evidence of this molecular mechanism as causative of FHM1 and expand the phenotypic spectrum of CACNA1A mutations with a child exhibiting severe SHM1 and non-episodic ataxia of congenital onset.

Opposite impacts of dietary versus supplemental calcium on cardiovascular health.

Commentary on: Li K, Kaaks R, Linseisen J, et al . Associations of dietary calcium intake and calcium supplementation with myocardial infarction and stroke risk and overall cardiovascular mortality in the Heidelberg cohort of the European prospective investigation into cancer and nutrition study (EPIC-Heidelberg). Heart 2012; 98 :920 - 5

The BH4 domain of Bcl-X(L) rescues astrocyte degeneration in amyotrophic lateral sclerosis by modulating intracellular calcium signals.

Collective evidence indicates that motor neuron degeneration in amyotrophic lateral sclerosis (ALS) is non-cell-autonomous and requires the interaction with the neighboring astrocytes. Recently, we reported that a subpopulation of spinal cord astrocytes degenerates in the microenvironment of motor neurons in the hSOD1(G93A) mouse model of ALS. Mechanistic studies in vitro identified a role for the excitatory amino acid glutamate in the gliodegenerative process via the activation of its inositol 1,4,5-triphosphate (IP(3))-generating metabotropic receptor 5 (mGluR5). Since non-physiological formation of IP(3) can prompt IP(3) receptor (IP(3)R)-mediated Ca(2+) release from the intracellular stores and trigger various forms of cell death, here we investigated the intracellular Ca(2+) signaling that occurs downstream of mGluR5 in hSOD1(G93A)-expressing astrocytes. Contrary to wild-type cells, stimulation of mGluR5 causes aberrant and persistent elevations of intracellular Ca(2+) concentrations ([Ca(2+)](i)) in the absence of spontaneous oscillations. The interaction of IP(3)Rs with the anti-apoptotic protein Bcl-X(L) was previously described to prevent cell death by modulating intracellular Ca(2+) signals. In mutant SOD1-expressing astrocytes, we found that the sole BH4 domain of Bcl-X(L), fused to the protein transduction domain of the HIV-1 TAT protein (TAT-BH4), is sufficient to restore sustained Ca(2+) oscillations and cell death resistance. Furthermore, chronic treatment of hSOD1(G93A) mice with the TAT-BH4 peptide reduces focal degeneration of astrocytes, slightly delays the onset of the disease and improves both motor performance and animal lifespan. Our results point at TAT-BH4 as a novel glioprotective agent with a therapeutic potential for ALS.

Changes in Aortic Pulse Wave Velocity in Hypertensive Postmenopausal Women: Comparison Between a Calcium Channel Blocker vs Angiotensin Receptor Blocker Regimen.

J Clin Hypertens (Greenwich). 2012;14:773-778. ©2012 Wiley Periodicals, Inc. Postmenopausal women are at greater risk for hypertension-related cardiovascular disease. Antihypertensive therapy may help alleviate arterial stiffness that represents a potential modifiable risk factor of hypertension. This randomized controlled study investigated the difference between an angiotensin receptor blocker and a calcium channel blocker in reducing arterial stiffness. Overall, 125 postmenopausal hypertensive women (age, 61.4±6 years; systolic blood pressure/diastolic blood pressure [SBP/DBP], 158±11/92±9 mm Hg) were randomized to valsartan 320 mg±hydrochlorothiazide (HCTZ) (n=63) or amlodipine 10 mg±HCTZ (n=62). The primary outcome was carotid-to-femoral pulse wave velocity (PWV) changes after 38 weeks of treatment. Both treatments lowered peripheral blood pressure (BP) (-22.9/-10.9 mm Hg for valsartan and -25.2/-11.7 mm Hg for amlodipine, P=not significant) and central BP (-15.7/-7.6 mm Hg for valsartan and -19.2/-10.3 mm Hg for amlodipine, P<.05 for central DBP). Both treatments similarly reduced the carotid-femoral PWV (-1.9 vs -1.7 m/s; P=not significant). Amlodipine was associated with a higher incidence of peripheral edema compared with the valsartan group (77% vs 14%, P<.001). BP lowering in postmenopausal women led to a reduction in arterial stiffness as assessed by PWV measurement. Both regimens reduced PWV to a similar degree after 38 weeks of treatment despite differences in central BP lowering, suggesting that the effect of valsartan on PWV is mediated through nonhemodynamic effects.

Production and Evaluation of Calcium Magnesium Acetate, HR-243, 1982

Calcium magnesium acetate (CMA) has been identified by Bjorksten Research Laboratories as an environmentally harmless alternative to sodium or calcium chloride for deicing highways. Their study found CMA to be noncorrosive to steel, aluminum and zinc with little or no anticipated environmental impact. When used, it degrades into elements found in abundance in nature. The deicing capabilities were found to be similar to sodium chloride. The neutralized CMA they produced did cause scaling of PC concrete, but they did not expect mildly alkaline CMA to have this effect. In the initial investigation of CMA at the Iowa DOT laboratory, it was found that CMA produced from hydrated lime and acetic acid was a light, fluffy material. It was recognized that a deicer in this form would be difficult to effectively distribute on highways without considerable wind loss. A process was developed to produce CMA in the presence of sand to increase particle weight. In this report the product of this process, which consists of sand particles coated with CMA, is referred to as "CMA deicer". The mixture of salts, calcium magnesium acetate, is referred to as "CMA". The major problems with CMA for deicing are: (1) it is not commercially available, (2) it is expensive with present production methods and (3) there is very little known about how it performs on highways under actual deicing conditions. In view of the potential benefits this material offers, it is highly desirable to find solutions or answers to these problems. This study provides information to advance that effort. The study consisted of four principal tasks which were: 1. Production of CMA Deicer The objective was to further develop the laboratory process for producing CMA deicer on a pilot plant basis and to produce a sufficient quantity for field trials. The original proposal called for producing 20 tons of CMA deicer. 2. Field Evaluation of CMA Deicer The objective was to evaluate the effectiveness of CMA deicer when used under field conditions and obtain information on application procedures. Performance was compared with a regular 50/50 mixture of sand and sodium chloride. 3. Investigation of Effects of CMA on PC Concrete The objective was to determine any scaling effect that mildly alkaline CMA might have on PC concrete. Comparison was made with calcium chloride. 4. Determine Feasibility of Producing High Magnesium CMA The objective was to investigate the possibility of producing a CMA deicer with magnesium acetate content well above that produced from dolomitic lime. A high magnesium acetate content is desirable because pure magnesium acetate has a water eutectic of -22 F° as compared with +5 F° for calcium acetate and is therefore a more effective deicer.

Aggregation of liposomes induced by calcium: A structural and kinetic study

In this work, the calcium-induced aggregation of phosphatidylserine liposomes is probed by means of the analysis of the kinetics of such process as well as the aggregate morphology. This novel characterization of liposome aggregation involves the use of static and dynamic light-scattering techniques to obtain kinetic exponents and fractal dimensions. For salt concentrations larger than 5 mM, a diffusion-limited aggregation regime is observed and the Brownian kernel properly describes the time evolution of the diffusion coefficient. For slow kinetics, a slightly modified multiple contact kernel is required. In any case, a time evolution model based on the numerical resolution of Smoluchowski's equation is proposed in order to establish a theoretical description for the aggregating system. Such a model provides an alternative procedure to determine the dimerization constant, which might supply valuable information about interaction mechanisms between phospholipid vesicles.