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1.
Phys Chem Chem Phys ; 13(8): 3033-8, 2011 Feb 28.
Artigo em Inglês | MEDLINE | ID: mdl-21135956

RESUMO

Over a range of hydration, unsaturated diacylglycerol/phosphatidylcholine mixtures adopt an inverse micellar cubic phase, of crystallographic space group Fd3m. In this study hydrated DOPC:DOG mixtures with a molar ratio close to 1 : 2 were examined as a function of hydrostatic pressure, using synchrotron X-ray diffraction. The small-angle diffraction pattern at atmospheric pressure was used to calculate 2-D sections through the electron density map. Pressure initially has very little effect on the structure of the Fd3m cubic phase, in contrast to its effect on hydrated inverse bicontinuous cubic phases. At close to 2 kbar, a sharp transition occurs from the Fd3m phase to a pair of coexisting phases, an inverse hexagonal H(II) phase plus an (ordered) lamellar phase. Upon increasing the pressure to 3 kbar, a further sharp transition occurs from the H(II) phase to a (fluid) lamellar phase, in coexistence with the ordered lamellar phase. These transitions are fully reversible, but show hysteresis. Remarkably, the lattice parameter of the Fd3m phase is practically independent of pressure. These results show that these two lipids are miscible at low pressure, adopting a single lyotropic phase (Fd3m); they then become immiscible above a critical pressure, phase separating into DOPC-rich and DOG-rich phases.


Assuntos
Diglicerídeos/química , Micelas , Fosfatidilcolinas/química , Pressão Hidrostática , Transição de Fase , Espalhamento a Baixo Ângulo , Temperatura , Difração de Raios X
2.
J Phys Condens Matter ; 18(28): S1105-24, 2006 Jul 19.
Artigo em Inglês | MEDLINE | ID: mdl-21690832

RESUMO

In recent years it has become evident that many biological functions and processes are associated with the adoption by cellular membranes of complex geometries, at least locally. In this paper, we initially discuss the range of self-assembled structures that lipids, the building blocks of biological membranes, may form, focusing specifically on the inverse lyotropic phases of negative interfacial mean curvature. We describe the roles of curvature elasticity and packing frustration in controlling the stability of these inverse phases, and the experimental determination of the spontaneous curvature and the curvature elastic parameters. We discuss how the lyotropic phase behaviour can be tuned by the addition of compounds such as long-chain alkanes, which can relieve packing frustration. The latter section of the paper elaborates further on the structure, geometric properties, and stability of the inverse bicontinuous cubic phases.

3.
Biochim Biophys Acta ; 1468(1-2): 41-54, 2000 Sep 29.
Artigo em Inglês | MEDLINE | ID: mdl-11018650

RESUMO

The excess water bilayer thickness, d(l,0), and molecular area, A(0), of lipid amphiphiles in the fluid lamellar phases of dioleoylphosphatidylcholine (DOPC) and dipalmitoleoylphosphatidylcholine (DPolPC) have been estimated between 15 and 50 degrees C and for dimyristoylphosphatidylcholine (DMPC) between 25 and 50 degrees C. These determinations have been made from X-ray measurements on samples of known water composition. With respect to temperature, T, d(l,0) and A(0) are well fitted to a linear equation. We find d(l,0) (A)=(35.68+/-0.02)-(0.0333+/-0.0006)T (degrees C) and A(0) (A(2))=(70.97+/-0.05)+(0.136+/-0.001)T (degrees C) for DOPC, d(l,0) (A)=(35.2+/-0.1)-(0.068+/-0.003)T (degrees C) and A(0) (A(2))=(59.7+/-0.2)+(0.210+/-0.006)T (degrees C) for DMPC, and d(l,0) (A)=(34.54+/-0.03)-(0.0531+/-0.0009)T (degrees C) and A(0) (A(2))=(67.12+/-0.09)+(0.173+/-0.003)T (degrees C) for DPolPC. The accuracy of these estimates depends largely on how accurately the excess water point is determined. Ideally, reliable X-ray and compositional data will be available around the excess water and it may be found by simple inspection, but this is the exception rather than the rule, since samples close to water excess normally sequester sizeable amounts of water in defects, which lead to an underestimate of d(l,0). and overestimate of A(0). In this paper, we report a methodology for identifying and removing such data points and fitting the remaining data in order to determine the excess water point.


Assuntos
Bicamadas Lipídicas/química , Água/análise , 1,2-Dipalmitoilfosfatidilcolina/química , Dimiristoilfosfatidilcolina/química , Liofilização , Fluidez de Membrana , Modelos Moleculares , Fosfatidilcolinas/química , Temperatura , Água/química , Difração de Raios X
4.
Biochim Biophys Acta ; 1327(1): 131-47, 1997 Jul 05.
Artigo em Inglês | MEDLINE | ID: mdl-9247174

RESUMO

The phase behaviour and structural parameters of a homologous series of saturated diacyl phosphatidylcholine/fatty acid 1:2 (mol/mol) mixtures having chain lengths from C12 to C20 were studied by X-ray diffraction and calorimetry, as a function of water content. The chain-melting transition temperatures of the 1:2 PC/FA mixtures are found to be largely independent of the degree of hydration. For all chain lengths, the tilted L(beta') and rippled P(beta') gel phases of the pure PC component are replaced by an untilted L(beta) gel phase in the 1:2 PC/FA mixtures. This gel phase swells considerably upon hydration, with a limiting water layer thickness in the range 18-24 A, depending on the chain length. However, unlike pure phospholipid systems, the lateral chain packing within the gel phase bilayers is essentially identical in both the dry and the fully hydrated states. The fluid bilayer L(alpha) phase is suppressed in the 1:2 mixtures, being replaced by inverse non-lamellar phases for all chain lengths greater than C12, and at all levels of hydration. For chain lengths of C16 and greater, the inverse hexagonal H(II) phase is formed directly upon chain melting, at all water contents. For the shorter chain length mixtures, the behaviour is more complex, with the H(II) phase forming at low hydration, but with bicontinuous cubic phases appearing at higher levels of hydration. The implications of these surprising results are explored, in terms of the effective hydrophilicity of the associated PC and FA headgroups and the packing within the interfacial region. We suggest that the presence of the fatty acids significantly alters the lateral stress profile across the lipid monolayer in the fluid state, compared to that of the corresponding pure PC system, such that inverse phases, where the interface bends towards the water, become strongly favoured. Furthermore, for short chain lengths, packing constraints favour the formation of phases with negative interfacial Gaussian curvature, such as the bicontinuous cubic phases, rather than the H(II) phase, which has more severe chain packing frustration.


Assuntos
Ácidos Graxos/química , Bicamadas Lipídicas/química , Fosfatidilcolinas/química , Calorimetria , Géis , Relação Estrutura-Atividade , Temperatura , Difração de Raios X
5.
Phys Rev E Stat Nonlin Soft Matter Phys ; 72(1 Pt 1): 011502, 2005 Jul.
Artigo em Inglês | MEDLINE | ID: mdl-16089965

RESUMO

This paper describes time-resolved x-ray diffraction data monitoring the transformation of one inverse bicontinuous cubic mesophase into another, in a hydrated lipid system. The first section of the paper describes a mechanism for the transformation that conserves the topology of the bilayer, based on the work of Charvolin and Sadoc, Fogden and Hyde, and Benedicto and O'Brien in this area. We show a pictorial representation of this mechanism, in terms of both the water channels and the lipid bilayer. The second section describes the experimental results obtained. The system under investigation was 2:1 lauric acid: dilauroylphosphatidylcholine at a hydration of 50% water by weight. A pressure-jump was used to induce a phase transition from the gyroid (Q(G)(II)) to the diamond (Q(D)(II)) bicontinuous cubic mesophase, which was monitored by time-resolved x-ray diffraction. The lattice parameter of both mesophases was found to decrease slightly throughout the transformation, but at the stage where the Q (D)(II) phase first appeared, the ratio of lattice parameters of the two phases was found to be approximately constant for all pressure-jump experiments. The value is consistent with a topology-preserving mechanism. However, the polydomain nature of our sample prevents us from confirming that the specific pathway is that described in the first section of the paper. Our data also reveal signals from two different intermediate structures, one of which we have identified as the inverse hexagonal (H(II)) mesophase. We suggest that it plays a role in the transfer of water during the transformation. The rate of the phase transition was found to increase with both temperature and pressure-jump amplitude, and its time scale varied from the order of seconds to minutes, depending on the conditions employed.

6.
Biochem Soc Symp ; (68): 27-33, 2001.
Artigo em Inglês | MEDLINE | ID: mdl-11573345

RESUMO

The folding mechanisms of integral membrane proteins have largely eluded detailed study. This is owing to the inherent difficulties in folding these hydrophobic proteins in vitro, which, in turn, reflects the often apparently insurmountable problem of mimicking the natural membrane bilayer with lipid or detergent mixtures. There is, however, a large body of information on lipid properties and, in particular, on phosphatidylcholine and phosphatidylethanolamine lipids, which are common to many biological membranes. We have exploited this knowledge to develop efficient in vitro lipid-bilayer folding systems for the membrane protein, bacteriorhodopsin. Furthermore, we have shown that a rate-limiting apoprotein folding step and the overall folding efficiency appear to be controlled by particular properties of the lipid bilayer. The properties of interest are the stored curvature elastic energy within the bilayer, and the lateral pressure that the lipid chains exert on the their neighbouring folding proteins. These are generic properties of the bilayer that can be achieved with simple mixtures of biological lipids, and are not specific to the lipids studied here. These bilayer properties also seem to be important in modulating the function of several membrane proteins, as well as the function of membranes in vivo. Thus, it seems likely that careful manipulations of lipid properties will shed light on the forces that drive membrane protein folding, and will aid the development of bilayer folding systems for other membrane proteins.


Assuntos
Proteínas de Membrana/química , Dobramento de Proteína , Bacteriorodopsinas/química , Técnicas In Vitro , Cinética , Bicamadas Lipídicas/química
7.
Phys Rev E Stat Nonlin Soft Matter Phys ; 65(3 Pt 1): 031501, 2002 Mar.
Artigo em Inglês | MEDLINE | ID: mdl-11909060

RESUMO

The time-dependent response of a surfactant hexagonal phase of a sodium dodecyl sulphate/pentanol/cyclohexane/brine system to stepped strain is investigated. The dynamics of the system is found to be governed by strain- and noise-induced yielding of the domains of the system. The effects of the applied strain magnitude and the ionic strength of the brine on the character of the transitions experienced by the system are reported.

8.
Rev Sci Instrum ; 84(8): 085104, 2013 Aug.
Artigo em Inglês | MEDLINE | ID: mdl-24007104

RESUMO

An automated laboratory based X-ray beamline with a multi-capillary sample chamber capable of undertaking small angle X-ray scattering measurements on a maximum of 104 samples at a time as a function of temperature between 5 and 85 °C has been developed. The modular format of the system enables the user to simultaneously equilibrate samples at eight different temperatures with an accuracy of ±0.005 °C. This system couples a rotating anode generator and 2D optoelectronic detector with Franks X-ray optics, leading to typical exposure times of less than 5 min for lyotropic liquid crystalline samples. Beamline control including sample exchange and data acquisition has been fully automated via a custom designed LabVIEW framework.


Assuntos
Espalhamento a Baixo Ângulo , Difração de Raios X/instrumentação , Automação , Cristais Líquidos , Temperatura
9.
Biochem Soc Trans ; 35(Pt 3): 498-501, 2007 Jun.
Artigo em Inglês | MEDLINE | ID: mdl-17511638

RESUMO

The nature of the bilayer motif coupled with the ability of lipids and proteins to diffuse freely through this structure is crucial to the viability of cells and their ability to compartmentalize domains contained therein. It seems surprising to find then that biological as well as model membranes exist in a dynamic state of mechanical stress. The stresses within such membranes are surprisingly large, typically reaching up to 50 atm (1 atm=101.325 kPa) at the core of the membrane and vary as a function of depth. The uneven distribution of lateral pressures within monolayer leaflets causes them to bend away from or towards the water interface. This can result in the formation of complex, self-assembled mesophases, many of which occur in vivo. Our knowledge of the principles underlying membrane mechanics has reached the point where we are now able to manipulate them and create nano-structures with reasonable predictability. In addition, they can be used both to explain and control the partitioning of amphipathic proteins on to membranes. The dependence of the dynamics of membrane-bound proteins and the chemical reactivity of amphipathic drug molecules on membrane stresses suggests that Nature itself takes advantage of this. Understanding and manipulating these internal forces will be a key element in creating self-assembled, biocompatible, nanoscale cell-like systems.


Assuntos
Membranas/química , Materiais Biocompatíveis/química , Bicamadas Lipídicas/química , Proteínas de Membrana/química , Membranas Artificiais , Modelos Biológicos , Nanoestruturas/química , Fosfatidilcolinas/biossíntese , Fosfatidilcolinas/química , Estresse Mecânico , Termodinâmica
10.
Biochemistry ; 38(29): 9328-36, 1999 Jul 20.
Artigo em Inglês | MEDLINE | ID: mdl-10413507

RESUMO

Three different lipid systems have been developed to investigate the effect of physicochemical forces within the lipid bilayer on the folding of the integral membrane protein bacteriorhodopsin. Each system consists of lipid vesicles containing two lipid species, one with phosphatidylcholine and the other with phosphatidylethanolamine headgroups, but the same hydrocarbon chains: either L-alpha-1, 2-dioleoyl, L-alpha-1,2-dipalmitoleoyl, or L-alpha-1,2-dimyristoyl. Increasing the mole fraction of the phosphatidylethanolamine lipid increases the desire of each monolayer leaflet in the bilayer to curve toward water. This increases the torque tension of such monolayers, when they are constrained to remain flat in the vesicle bilayer. Consequently, the lateral pressure in the hydrocarbon chain region increases, and we have used excimer fluorescence from pyrene-labeled phosphatidylcholine lipids to probe these pressure changes. We show that bacteriorhodopsin regenerates to about 95% yield in vesicles of 100% phosphatidylcholine. The regeneration yield decreases as the mole fraction of the corresponding phosphatidylethanolamine component is increased. The decrease in yield correlates with the increase in lateral pressure which the lipid chains exert on the refolding protein. We suggest that the increase in lipid chain pressure either hinders insertion of the denatured state of bacterioopsin into the bilayer or slows a folding step within the bilayer, to the extent that an intermediate involved in bacteriorhodopsin regeneration is effectively trapped.


Assuntos
Bacteriorodopsinas/química , Bicamadas Lipídicas/química , Proteínas de Membrana/química , Dobramento de Proteína , Bacteriorodopsinas/metabolismo , Ácidos Cólicos/química , Dimiristoilfosfatidilcolina/química , Halobacterium salinarum/química , Bicamadas Lipídicas/metabolismo , Proteínas de Membrana/metabolismo , Micelas , Modelos Moleculares , Fosfatidilcolinas/química , Fosfatidiletanolaminas/química , Éteres Fosfolipídicos/química , Pressão , Estrutura Secundária de Proteína , Espectrofotometria Ultravioleta
11.
J Fluoresc ; 3(3): 149-52, 1993 Sep.
Artigo em Inglês | MEDLINE | ID: mdl-24234824

RESUMO

We have used dipyrenylphosphatidylcholines (dipyPCs) to study the pressure in the fluid lamellar phase formed by mixtures of fully hydrated dioleoylphosphatidylcholine (DOPC) and dioleoylphosphatidylethanolamine (DOPE). As we increase the DOPE mole fraction at 25°C we observe a linear increase in the ratio of the excimer-to-monomer signal (E/M1). We argue that this observation can be understood in terms of an increase in the lateral pressure in the chain region, i.e., in the bilayer plane. This change itself is driven by the decrease in lateral pressure between headgroups as we add DOPE. We expect the lateral pressure to vary in magnitude as we probe the bilayer at different depths [1]. We have confirmed this by recording E/M1 using di[10-(pyren-1-yl)decanoyl]phosphatidylcholine (10dipyPC) and di[4-(pyren-1-yl)butanoyl]phosphatidylcholine (4dipyPC). We find that in 100% DOPC the E/M1 for 4dipyPC is 2.5 times greater than that for 10dipyPC. The above observations can all be rationalized in terms of changes in the lateral pressure profile. An inverse hexagonal liquid crystalline phase is found in the range 100-83% DOPE [2]. In this region of the phase diagram we observe a quadratic variation in E/M1, with a minimum at 95% DOPE. We hypothesize that this variation reflects the chain stretching that is necessitated by the geometrical packing constraints of the hexagonal phase [3]. Again, we find that the E/M1 for 4dipyPC is greater than that for 10dipyPC, but in this phase only by a factor of two.

12.
Faraday Discuss ; (111): 41-53; discussion 69-78, 1998.
Artigo em Inglês | MEDLINE | ID: mdl-10822599

RESUMO

In this work we present data from a homologous series of di-pyrenyl phosphatidylcholine (dipyPC) probes which can sense lateral pressure variations in the chain region of the amphiphilic membrane (lateral pressures are tangential to the interface). The dipyPC has pyrene moieties attached to the ends of equal length acyl chains on a phosphatidylcholine molecule. Ultraviolet stimulation produces both monomer and excimer fluorescence from pyrene. At low dilutions of dipyPC in model membranes the excimer signal is entirely intra-molecular and since it depends on the frequency with which the pyrene moieties are brought into close proximity, the relative intensity of the excimer to monomer signal, eta, is a measure of the pressure. We synthesised or purchased dipyPC probes with the pyrene moieties attached to acyl chains having 4, 6, 8 and 10 carbon atoms and then measured eta in fully hydrated bilayers composed of dioleoylphosphatidylcholine and dioleoylphosphatidylethanolamine (DOPC and DOPE respectively). Although the resolution of our measurements of lateral pressure as a function of distance into the monolayer was limited, we did observe a dip in the excimer signal in the region of the DOPC/DOPE cis double bond. As we isothermally increased the DOPE composition, and hence the desire for interfacial curvature, we observed, as expected, that the net excimer signal increased. However this net increase was apparently brought about by a transfer of pressure from the region around the glycerol backbone to the region near the chain ends, with the lateral pressure dropping above the cis double bond but increasing at a greater rate beyond the double bond.


Assuntos
Membranas Artificiais , Fosfatidilcolinas , Fosfatidiletanolaminas , Propriedades de Superfície
13.
Proc Natl Acad Sci U S A ; 97(16): 9032-6, 2000 Aug 01.
Artigo em Inglês | MEDLINE | ID: mdl-10908674

RESUMO

The activity of CTP:phosphocholine cytidylyltransferase, a rate-limiting enzyme in phosphatidylcholine biosynthesis, is modulated by its interaction with lipid bilayers [Kent, C. (1997) Biochim. Biophys. Acta 1348, 79-90]. Its regulation is of central importance in the maintenance of membrane lipid homeostasis. Here we show evidence that the stored curvature elastic stress in the lipid membrane's monolayer modulates the activity of CTP:phosphocholine cytidylyltransferase. Our results show how a purely physical feedback signal could play a key role in the control of membrane lipid synthesis.


Assuntos
Colina-Fosfato Citidililtransferase/metabolismo , Fosfolipídeos/metabolismo , Membrana Celular/enzimologia , Elasticidade , Bicamadas Lipídicas
14.
Biochem Soc Trans ; 29(Pt 4): 408-13, 2001 Aug.
Artigo em Inglês | MEDLINE | ID: mdl-11497998

RESUMO

Protein folding has been at the forefront of molecular cell biology research for several years. However, integral membrane proteins have eluded detailed molecular level study until recently. One reason is the often apparently insurmountable problem of mimicking the natural membrane bilayer with lipid or detergent mixtures. There is nevertheless a large body of information on lipid properties and in particular on phosphatidylcholine and phosphatidylethanolamine lipids, which are common to many biological membranes. We have exploited this knowledge to design efficient in vitro, lipid-bilayer folding systems for membrane proteins. Bacteriorhodopsin has been used as a model system for our initial studies: we have shown that a rate-limiting apoprotein folding step and the overall folding efficiency seem to be controlled by particular properties of the lipid bilayer. The properties of interest are the stored curvature elastic energy within the bilayer and the lateral pressure that the lipid chains exert on their neighbouring folding protein. These are generic properties of the bilayer that can be achieved with simple mixtures of many types of biological lipid and seem to be important in vivo.


Assuntos
Proteínas de Membrana/metabolismo , Dobramento de Proteína , Proteínas de Bactérias/química , Bacteriorodopsinas/química , Bacteriorodopsinas/metabolismo , Bicamadas Lipídicas/química , Proteínas de Membrana/química , Complexo de Proteínas do Centro de Reação Fotossintética/química , Desnaturação Proteica , Estrutura Secundária de Proteína
15.
Crit Rev Biochem Mol Biol ; 36(6): 501-603, 2001.
Artigo em Inglês | MEDLINE | ID: mdl-11798093

RESUMO

The study of membrane protein folding is a new and challenging research field. Consequently, there are few direct studies on the in vitro folding of membrane proteins. This review covers work aimed at understanding folding mechanisms and the intermolecular forces that drive the folding of integral membrane proteins. We discuss the kinetic and thermodynamic studies that have been undertaken. Our review also draws on closely related research, mainly from purification studies of functional membrane proteins, and gives an overview of some of the successful methods. A brief survey is also given of the large body of mutagenesis and fragment work on membrane proteins, as this too has relevance to the folding problem. It is noticeable that the choice of solubilizing detergents and lipids can determine the success of the method, and indeed it appears that particular lipid properties can be used to control the rate and efficiency of folding. This has important ramifications for much in vitro folding work in that it aids our understanding of how to obtain and handle folded, functional protein. With this in mind, we also cover some relevant properties of model, lipid-bilayer systems.


Assuntos
Proteínas de Membrana/química , Proteínas de Membrana/metabolismo , Dobramento de Proteína , Proteínas de Bactérias/química , Proteínas de Bactérias/metabolismo , Membrana Celular/química , Membrana Celular/metabolismo , Cinética , Lipídeos de Membrana/metabolismo , Conformação Proteica , Desnaturação Proteica , Engenharia de Proteínas , Renaturação Proteica , Termodinâmica
16.
Biochemistry ; 36(1): 197-203, 1997 Jan 07.
Artigo em Inglês | MEDLINE | ID: mdl-8993334

RESUMO

The regeneration kinetics of the integral membrane protein bacteriorhodopsin have been investigated in a lipid-based refolding system. Previous studies on bacteriorhodopsin regeneration have involved detergent-based systems, and in particular mixed dimyristoylphosphatidylcholine (DMPC)/CHAPS micelles. Here, we show that the short chain lipid dihexanoylphosphatidylcholine (DHPC) can be substituted for the detergent CHAPS and that bacteriorhodopsin can be regenerated to high yield in mixed DMPC/DHPC micelles. Bacteriorhodopsin refolding kinetics are measured in the mixed DMPC/DHPC micelles. Rapid, stopped flow mixing is employed to initiate refolding of denatured bacterioopsin in SDS micelles with mixed DMPC/DHPC micelles and time-resolved fluorescence spectroscopy to follow changes in protein fluorescence during folding. Essentially identical refolding kinetics are observed for mixed DMPC/CHAPS and mixed DMPC/DHPC micelles. Only one second-order retinal/apoprotein reaction is identified, in which retinal binds to a partially folded apoprotein intermediate, and the free energy of this retinal binding reaction is found to be the same in both types of mixed micelles. Formation of the partially folded apoprotein intermediate is a rate-limiting step in protein folding and appears to be biexponential. Both apparent rate constants are found to be dependent on the relative proportion of DMPC present in the mixed DMPC/DHPC micelles as well as on the pH of the aqueous phase. Increasing the DMPC concentration should increase the bending rigidity of the amphiphilic bilayer, and this is found to slow the rate of formation of the partially folded apoprotein intermediate. Increasing the mole fraction of DMPC from 0.3 to 0.6 slows the two apparent rate constants associated with formation of this intermediate from 0.29 and 0.031 to 0.11 and 0.013 s-1, respectively. Formation of the intermediate also slows with increasing pH, from 0.11 and 0.013 s-1 at pH 6 to 0.033 and 0.0053 s-1 at pH 8. Since this pH change has no known effect on the phase behavior of lecithins, this is more likely to represent a direct effect on the protein itself. Thus, it appears to be possible to control the rate-limiting process in bacterioopsin folding through both bilayer bending rigidity and pH.


Assuntos
Bacteriorodopsinas/metabolismo , Bicamadas Lipídicas/metabolismo , Proteínas de Membrana/química , Dobramento de Proteína , Ácidos Cólicos/farmacologia , Dimiristoilfosfatidilcolina/farmacologia , Concentração de Íons de Hidrogênio , Cinética , Bicamadas Lipídicas/farmacologia , Lipossomos/metabolismo , Modelos Químicos , Éteres Fosfolipídicos/farmacologia , Retinaldeído/metabolismo , Retinaldeído/farmacologia , Espectrometria de Fluorescência , Espectrofotometria , Termodinâmica
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