Francesca Bonomini

Francesca Filippini

Tony Hayek

Michael Aviram

Shlomo Keidar

Luigi F Rodella

Raymond Coleman

Rita Rezzani
email: rezzani@med.unibs.it

Apolipoprotein E and its role in aging and survival

Keywords: Apolipoprotein E, aging, liver, kidney, heart

The study of biological aging has seen spectacular progress in the last decade and markers are increasingly employed for understanding physiological processes that change with age. Recently, it has been demonstrated that apolipoprotein E (apoE) has a major impact on longevity, but its mechanisms are still not fully understood. ApoE-deficient (E o ) mice have proved to be a very popular model for studying spontaneous hypercholesterolemia and the subsequent development of atherosclerotic lesions, but only limited data are available with regard to aging and aging changes.

We used this murine model to better characterize the involvement of apoE in aging and to evaluate its role in the maintenance of normal organ morphology. Our results show that E 0 mice at different ages (6, 12, 20 weeks old) developed age-dependent morphological and biochemical alterations, including fibrosis (newly formed collagen), pro-inflammatory cytokine (IL-6 and iNOS), lipofuscin accumulation, and decrease of antioxidant enzymes (superoxide dismutase and catalase) in several organs (kidney, liver and heart). It is significant that the observed degenerative findings in E 0 mice at different ages (6, 12, 20 weeks old) were not identified in control mice (C57BL), at 6, 12 and 20 weeks of age. Consequently, since these mice showed enzymatic and structural alterations, normally linked to the age, such as increase of lipofuscin, pro-inflammatory cytokines and decrease of antioxidant enzymes, we can conclude that apoE is a useful player in studies of longevity and age-related diseases, such as inflammatory status and atherosclerosis that are known risk factors for functional decline and early mortality. Moreover, it is possible that apoE may also play a role in other pathological conditions including, for example, cancer, rheumatoid arthritis and macular degeneration.

Introduction

Aging is a multifactorial process and attempts at understanding the fundamental causes of aging are limited by the complexity of the problem [START_REF] Kirkwood | Understanding the odd science of aging[END_REF][START_REF] Rajawat | Aging: central role for autophagy and the lysosomal degradative system[END_REF]. However, studying aging processes at the cellular and molecular levels is complicated and subjected to confounding experimental and environmental factors [START_REF] Dice | Cellular and molecular mechanisms of aging[END_REF]. The lack of precise, well defined and reliable cellular and biochemical markers of aging has hindered efforts to identify the primary mechanisms and separate them from secondary effects. Many theories have been proposed about the causes of aging [START_REF] Weinert | Theories of aging[END_REF] but broadly these can be divided into two main categories [START_REF] Troen | The biology of aging[END_REF]: the stochastic theories and the genetic theories. The fundamental concept behind the stochastic theories is the progressive accumulation of "damage" that occurs throughout the entire lifespan of cells and tissue [START_REF] Rattan | Principles and practice of hormetic treatment of aging and age-related diseases[END_REF]. Such damage may result from toxic byproducts of routine metabolism or inefficient repair and defensive systems. It is apparent that in long-lived post-mitotic cells, such as neurons and cardiomyocytes, accumulation of lesions may prove more critical and detrimental. The genetic theories consider aging as part of the genetically programmed and controlled events during the lifespan that includes periods of development, maturation, senescence and death [START_REF] Finch | Genetics of aging[END_REF]. The genetic and stochastic theories, however, are not mutually exclusive. Since the aim of this current study has been to consider the contribution of stochastic events in the aging process, further discussion of the genetic theories is not considered.

The stochastic theories speculate that cumulative damage induced by random environmental factors or insults at various levels of an organism represent the main causes of aging. These include damage to DNA or inability to repair, damage to cell organelles and tissues by free radicals of oxygen, protein cross-linking and the limited ability of the cellular turnover mechanisms to remove such damaged structures and molecules. The progressive accumulation of such damage at a level that inhibits or disrupts normal physiological function may lead to progressive aging changes and senescence [START_REF] Rajawat | Aging: central role for autophagy and the lysosomal degradative system[END_REF]. Moreover, it has been demonstrated that free radicals or reactive oxygen species (ROS) accumulate over time causing progressive deterioration in the function of cells, tissues and organ systems, which may ultimately cause the death of cells and damage to a system. An excessive increase in ROS production has been implicated in the pathogenesis of many disorders including: atherosclerosis, cardiovascular diseases, hypertension, ischemia/reperfusion injury, diabetes mellitus, neurodegenerative and immuno-inflammatory diseases [START_REF] Roberts | Oxidative stress and metabolic syndrome[END_REF]. Moreover, ROS are highly reactive and may modify and inactivate proteins, lipids, DNA, and RNA and induce cellular dysfunctions. To prevent free radical-induced cellular damage, the organism has developed a defense mechanism, the antioxidative system. This system includes antioxidant enzymes such as superoxide dismutase (SOD) and catalase (CAT) [START_REF] Melov | Therapeutics against mitochondrial oxidative stress in animal models of aging[END_REF].

In order to detect the age-related changes in the biochemical composition of tissues, many markers of aging have been described, for example an increased accumulation of lipofuscin (age pigment) [START_REF] Florini | Effect of rat age on serum levels of growth hormone and somatomedins[END_REF], increased cross-linking in molecules of the extracellular matrix such as collagen [START_REF] Bjorksten | Cross linkage and the aging process[END_REF] and alteration in pro-inflammatory cytokines (Chung et al., 2002).

Accumulated data strongly suggest that continuous (chronic) upregulation of pro-inflammatory mediators (e.g. IL-6, iNOS) are induced during the aging process due to an age-related redox imbalance that activates many pro-inflammatory signaling pathways, including the NF-B signaling pathway [START_REF] Chung | Molecular inflammation: underpinnings of aging and age-related diseases[END_REF]. Other notable age-related changes involve alterations in the rate of transcription of specific genes, the rate of protein synthesis and turnover and numerous posttranslational protein modifications, such as glycation and oxidation [START_REF] Schneider El | Handbook of the Biology of Aging[END_REF].

These alterations with aging presumably lead to a diminished ability to maintain homeostasis and a subsequent reduced ability to adaptively respond to environmental stimuli or insults [START_REF] Adelman | Endocrine regulation of enzyme activity in aging animals of different genotypes[END_REF].

Biomarkers are increasingly employed in empirical studies of human populations to understand physiological processes that change with age, including diseases whose onset appears linked to age, and the aging process itself [START_REF] Crimmins | Biomarkers related to aging in human populations[END_REF]. Recently, it has been demonstrated that apolipoprotein E (apoE) has a large impact on longevity [START_REF] Smith | Apolipoproteins and aging: emerging mechanisms[END_REF][START_REF] Ang | Apolipoprotein E, an important player in longevity and age-related diseases[END_REF] even if the mechanisms involved in aging are not fully understood. In addition, apoE is well known to play a major role in lipid homeostasis, particularly in determination of the levels of low density lipoprotein (LDL) and high density lipoprotein (HDL) cholesterol, which are directly and inversely correlated with risk for cardiovascular disease, respectively [START_REF] Smith | Apolipoproteins and aging: emerging mechanisms[END_REF].

Numerous murine models are available for studying the aging process. Most of these models are based on known mutations that cause progeroid disease in humans or are involved in DNA repair and cell senescence [START_REF] Ang | Apolipoprotein E, an important player in longevity and age-related diseases[END_REF]. In particular, apoE-deficient (E o ) mice, since their introduction in the early 1990s, have proved to be a very popular model for studying spontaneous hypercholesterolemia and the subsequent development of atherosclerotic lesions [START_REF] Coleman | A mouse model for human atherosclerosis: long-term histopathological study of lesion development in the aortic arch of apolipoprotein E-deficient (E0) mice[END_REF], however, only limited data are reported in regard to successful aging. [START_REF] Moghadasian | Pathophysiology of apolipoprotein E deficiency in mice: relevance to apo E-related disorders in humans[END_REF] observed that 35% of E o mice died suddenly after 18 months, whereas [START_REF] Ang | Apolipoprotein E, an important player in longevity and age-related diseases[END_REF] showed that on a high fat diet, these mice, even after 7 months of age, exhibited major pathological disorders including: cachexia, severe cutaneous xanthomas, dramatic loss of subcutaneous fat and low body mass, kyphosis and hypokinesis.

On the basis of the above premises, and in order to evaluate the role of apoE in aging processes, we performed a morphological and biological study on apolipoprotein E knock-out mice at different ages aged 6 to 20 weeks at sacrifice to characterize better the spontaneous changes in key organs such as kidney, liver and heart of E o mice. So, we decided to use hematoxylin-eosin and periodic acid Schiff staining for morphological study and biological markers, such as IL-6, iNOS, SOD and catalase since, as above reported, are a useful tool for aging damage.

Materials and methods

Apolipoprotein E-deficient (E o ) mice E o mice are widely used as animal models for atherosclerosis, since they develop severe hypercholesterolemia on a regular chow diet and the histopathology of their lesions closely resemble those in humans [START_REF] Plump | Severe hypercholesterolemia and atherosclerosis in apolipoprotein E-deficient mice created by homologous recombination in ES cells[END_REF][START_REF] Meir | Atherosclerosis in the apolipoprotein-E-deficient mouse: a decade of progress[END_REF]. The mice stock (provided courtesy of Prof. Jan Breslow, Rockefeller University, NY) were bred and housed in the Animal Care Facility of the Rappaport Faculty of Medicine, and the protocols for the study were approved by the Committee for Supervision of Animal Experiments of the Technion-Israel Institute of Technology. Female E° mice were divided into six groups: group I (control, 6 weeks old at sacrifice); group II (control, 12 weeks old); group III (control, 20 weeks old); group IV (E o mice, 6 weeks old); group V (E o mice, 12 weeks old); group VI (E o mice, 20 weeks old). The age-matched control mice were C57BL6 and not deficient in apolipoprotein E. Mice were anesthetized with ether using a nasal cone and hearts, kidney, and liver were rapidly dissected, washed briefly in phosphate buffer (0.1M, pH 7.4) and immersion-fixed in 10% neutral buffered formalin, prior to embedding in paraffin wax using routine procedures. 5µm thick serial sections were cut using a microtome. For histopathological evaluation, sections were stained with hematoxylin-eosin, picrosirius red and periodic acid Schiff staining. These latters have been made according to [START_REF] Junqueira | Picrosirius staining plus polarization microscopy, a specific method for collagen detection in tissue sections[END_REF] and [START_REF] Stacchiotti | Stress proteins expression in rat kidney and liver chronically exposed to aluminium sulphate[END_REF] respectively. Regarding picrosirius red staining, under polarized light microscopy, the type III collagen (constitutive) appeared green, while type I collagen (newly formed) yellow/red [START_REF] Rizzoni | Changes in extracellular matrix in subcutaneous small resistance arteries of patients with primary aldosteronism[END_REF].

Immunohistochemical localization

For the immunohistochemical study, sections were deparaffinized and rehydrated by routine protocol and immersed in 3% hydrogen peroxide (H 2 O 2 ) in methanol for 30 min to quench endogenous peroxidase. Sections obtained from all groups were incubated for 1 h, side by side, at room temperature, in normal serum (diluted 1:5 Dako Cytomation, Glostrup, Denmark). The samples were then incubated with primary antibodies directed against the specific antigens (4 and IL-6, SOD, CAT and iNOS, Santa Cruz Biotechnology, Santa Cruz, CA, USA), diluted 1:100 in Tris buffer (0.1M, pH7.4; TBS) overnight. Sections were washed in TBS, then incubated with a biotinylated secondary antibody diluted 1:50 for 1h, and then with the avidin-biotin horseradish peroxidase complex (ABC Kit; Dakopatts, Milan, Italy), prepared according to manufacturer's instructions for a further 1hr. Sections were immersed in a solution of 0.05% 3.3-diamino-benzidine tetrahydrochloride (DAB) and 0.03% H 2 O 2 for 10 min, then counterstained with hematoxylin, dehydrated, cleared and mounted according to routine protocols.

Control reactions were performed in the absence of primary antibody and in the presence of isotype-matched IgGs. For quantitative analysis, all samples were evaluated at 400 x magnification using an Olympus BX50 optical microscope, (Olympus, Hamburg, Germany), equipped with an image analysis program (Image-Pro Plus TM , Media Cybernetics, Silver Spring, MD). The integrated optical density (IOD), resulting from DAB staining intensity, was calculated for arbitrary zones in 10 fields of standard area (0.04 mm 2 / field) per slide. The DAB staining intensity was calculated using monochromatic (grey) light and the controls were subuntracted. Five sections were analyzed for each animal. The data obtained from the different experimental groups were analyzed and scored blindly. Statistical significance of differences between the experimental groups was estimated using ANOVA and Bonferroni tests with p<0.05 considered significant.

Measurement of morphological damage

For quantitative analysis of morphological damage, visualized by hematoxylin-eosin, the percentage of glomerulosclerosis, tubular number and percentage of polyploid hepatocytes were calculated in 10 fields for each sample by an independent observer, using an optical polarized light microscope at a final magnification of 200 x . Statistical significance of differences between the experimental groups was estimated using ANOVA and Bonferroni testes with p<0.05 considered significant.

For quantitative analysis of morphological damage, visualized by picrosirius red and periodic acid Schiff staining, digital pictures at 40X were taken using an optical microscope (Olympus, Germany) and then analyzed using an image program (Image-Pro Plus TM 4.5.1, Milan, Italy) in order to calculate the percentage of stained area. The yellow/red and green fibers, for picrosirius staining, and the pink lipofuscins, for periodic acid Schiff, were meaasured counting ten randomly collected fields of identical area (0.86 mm 2 ) for each section. The data were pooled to represent a mean value and a statistical analysis. Statistical significance of differences between the experimental groups was estimated using ANOVA and Bonferroni tests with p<0.05 considered significant.

Results

Histology and immunohistochemistry in kidney, liver and heart

Kidney

Kidneys stained by the picrosirius Red method, both cortex and medulla showed very little collagen, distributed throughout anatomical structures. Under a polarized light microscope, these collagen fibers appeared as a green colour indicating the presence of constitutive collagen (Fig. 1A).

Hematoxylin-eosin staining from control animals (6, 12 and 20 weeks old) showed a similar morphology with normal cytoarchitecture (Fig. 1A, inset).

On the contrary, E o (6, 12, 20 week old) mice showed interstitial fibrosis, which was more evident in the 20 week old mice (Fig. 1B) and glomerulosclerosis, tubular atrophy (Fig. 1B, inset). Marked interstitial fibrosis was seen both around tubules and glomeruli in the cortex and around vasa recta in the medulla. The extracellular matrix deposition around glomeruli, tubular structures, and vasa recta were observed under polarized light microscope; this fibrosis showed mainly newly formed collagen indicated by yellow/red fibers. The age-dependent quantitative data are reported in Fig. 1C.

The tubule number decreases with age (Fig. 2A) whereas tubular atrophy increases. The renal morphology is showed in the insets of Figs. 2A and2B. In addition, the percentage of glomeruli showing global glomerulosclerosis increases in E° mice from 6 to 20 weeks and it was higher in the subcapsular cortex compared to juxtamedullary zone (Fig. 2B). In detail, the cortical changes consisted in tubular and glomerular injury with dilation of the lumen and Bowman' s capsule, respectively. Alterations of the medulla showed vasa recta disruption.

IL-6 and iNOS immunostaining showed a similar positivity in the renal cortex and medulla. In detail, IL-6 immunostaining displayed very weak positivity in control animals at different ages (Fig. 3A) and moderate staining in E o mice aged 6 weeks. With aging the positivity increased for both the proteins and in E o mice (20 weeks old), it become strong (Fig. 3B). In particular, IL-6 and iNOS immunostaining was localized in epithelial cells of tubules.

No iNOS increase and no antioxidant enzyme decrease were found in control animals at different age (Figs. 4A,4B). As regards the antioxidant enzyme (SOD and catalase) analysis, we observed a decrease of their positivity already in 6 week old E° animals; the positivity ranged from strong to moderate in animals 20 weeks old (Figs. 4B), both in the cortex and medulla.

Liver

The control animals at different age showed normal cytoarchitecture of hepatic parenchyma. On the contrary, with aging in E° mice, the liver showed evident morphological alterations, including disorganization of hepatic parenchyma with widespread cell swelling and congestion of sinusoids, with a significant increase of fatty deposits in hepatocytes. We observed an accumulation of lipofuscin (red stained with periodic acid Schiff method according to [START_REF] Brunk | Lipofuscin: mechanisms of age-related accumulation and influence on cell function[END_REF] within the hepatocytes linked to age in comparison with age-matched control mice (Fig. 5A,5B). This accumulation was very high in 20 week old E° mice. The quantitative data regarding lipofuscin accumulation are reported in Fig. 5C. 20 week old E o mice liver also showed polyploid hepatocytes and the percentage of these increased by 40% when compared to the control animals (Fig. 6).

Picrosirius red staining demonstrated limited perisinusoidal fibrosis in 6 week old E° mice, whereas such fibrosis was well evident in 20 week old E° mice. Polarized microscopy showed few perisinusoidal yellow/red fibers of collagen in young animals (E° mice), whereas more fibers of this collagen were evident in 20 week old mice (data not shown). No fibrosis was evident in control animals at different age (data not shown).

IL-6 and iNOS immunostaining, evident in cytoplasm of the hepatocytes, were scarce in liver of control animals at different age, whereas it increased with age in E o mice (Figs. 7A,7B). As depicted in Figs. 4A,4B, the obtained results for antioxidant enzymes were similar to those observed in the kidney.

Heart

The connective tissue in heart of control animals at different age was thin, well highlighted and green stained under light polarized microscope after Picrosirius Red method (Fig. 8A). The parenchyma had a normal morphology with abundant cardiomyocytes and scarce connective tissue (Fig. 8A, inset). Degenerative changes were evident in E o mice in a time-dependent manner. The connective tissue was clearly increased and the fibrosis, under light polarized microscopy after Picrosirius Red method, was evident around these cardiomyocytes (Fig. 8B).The fibers still had abundant cytoplasm as in the control mice (Fig. 8B,inset). The age-dependent quantitative data are shown in Fig. 8C. Moreover, the fibrosis increase was associated with a decrease in the number of nuclei per myocytes and with accumulation of lipofuscin pigment (data not shown).

In 6 week old E o mice, cardiomyocytes showed a weak positivity for iNOS and IL-6. This immunohistochemical pattern changed with ageing showing an increase in immunostaining of these proteins with a strong positivity at 20 weeks of age, very different from control animals as shown in Figs.4A, 9A, 9B. Both of the proteins were immunolocalized in the sarcoplasm of cardiomyocytes.

As depicted in Fig. 4B, the results for the antioxidant enzymes were similar to those observed in the kidney. No variation in the studied markers was found in control animals at different ages.

Discussion

Our results showed that E° mice at different ages had age-dependent morphological and biochemical alterations, such as fibrosis, pro-inflammatory cytokine increase, lipofuscin accumulation and a decrease of antioxidant enzymes in different organs (kidney, liver and heart). It is significant that the observed observations in E° mice at different age (6, 12, 20 weeks old) were not identified in age-matched control animals.

The generation of E° mice has been one of the most critical advances in the elucidation of factors affecting atherogenesis [START_REF] Coleman | A mouse model for human atherosclerosis: long-term histopathological study of lesion development in the aortic arch of apolipoprotein E-deficient (E0) mice[END_REF][START_REF] Dursun | Proteomic profiling in apolipoprotein E-deficient mice during atherosclerosis progression[END_REF]. These mice have marked hypercholesterolemia because of deficiency of apolipoprotein E, develop spontaneous atherosclerotic plaques with many features similar to human lesions and allow investigation of atherosclerosis from its earliest stages [START_REF] Rosenblat | Pomegranate byproduct administration to apolipoprotein e-deficient mice attenuates atherosclerosis development as a result of decreased macrophage oxidative stress and reduced cellular uptake of oxidized low-density lipoprotein[END_REF]. It is currently the most popular murine model in cardiovascular research and has revealed important insights into atherosclerosis [START_REF] Aviram | Pomegranate phenolics from the peels, arils, and flowers are antiatherogenic: studies in vivo in atherosclerotic apolipoprotein e-deficient (E 0) mice and in vitro in cultured macrophages and lipoproteins[END_REF]. This mouse model provides the opportunity to analyze protein changes during various stages of atherogenesis under well-defined laboratory conditions and in animals with identical genetic background, facilitating proteomic comparisons by limiting biological variation [START_REF] Rodella | Atherosclerosis and the protective role played by different proteins in apolipoprotein E-deficient mice[END_REF]. Whereas the important role of apoE protein in atherosclerosis has been well characterized, its involvement in longevity and age-related diseases is still not resolved. Although many potential mechanisms, mainly in recent years, have been discussed, much more progress needs to be made, and researchers in this field are still debating whether increasing or decreasing apoE expression would be beneficial in aging [START_REF] Smith | Apolipoproteins and aging: emerging mechanisms[END_REF].

As far as we are aware, there are no morphological data regarding the cytoarchitecture of organs of E o mice and few biochemical and molecular findings showing changes in apoE expression linked to age. Our results showing damage in several organs (kidney, liver and heart) could be a useful tool for studying a progressive decline of different cellular functions related to aging.

Regarding renal structural changes that we have found, there are many papers reporting similar agerelated alterations. [START_REF] Zhou | The aging kidney[END_REF] showed that renal aging is associated with alterations in renal morphology and a decline in renal function, which is accelerated and/or accentuated by diseases such as diabetes mellitus and hypertension. The histological changes indicate glomerulosclerosis, tubular atrophy and interstitial fibrosis. These features are soundly correlated with aging-related intimal fibrosis of the interlobular arteries, a change that may be accelerated by hypertension and diabetes mellitus [START_REF] Zhou | The epidemiology of age-related eye diseases in Mainland China[END_REF]. In addition, the same authors, in agreement with other researchers [START_REF] Martin | Renal ageing[END_REF], showed that, in parallel with a reduction in the number of viable glomeruli, there is an increase in the number and percentage of glomeruli showing agerelated sclerosis. Subcapsular cortical glomeruli are particular susceptible to sclerosis in comparison to juxtamedullary region; this condition is generally thought to be a non-specific end-stage morphological change resulting from a large range of insults, including ischemia and a wide range of immunological disorders; however, different types of segmental sclerosing lesions have been clearly shown to result from different stimuli. These changes may be interpreted as an attempt to protect against adverse processes influencing the safety of the kidney and reducing its capacity to respond to later insults.

With regard to liver and heart morphology we found an accumulation of lipofuscin and this is a recognized hallmark of aging occurring with a rate inversely related to longevity [START_REF] Schmucker | Liver function and phase I drug metabolism in the elderly: a paradox[END_REF][START_REF] Terman | The aging myocardium: roles of mitochondrial damage and lysosomal degradation[END_REF]. Lipofuscin is not an inert material but may have a chemically very reactive surface that can disturb cellular metabolism. Lipofuscin inherently causes toxic effects, in part because of its ability to bind metals, such as iron, copper, zinc, manganese, and calcium, in a concentration up to 2%. One of the dramatic effects of lipofuscin is its ability to decrease proteasomal and lysosomal degradation. In using lysosomal enzyme capacity and lysosomal space, it reduces lysosomal degradation capacity [START_REF] Massey | Chaperone-mediated autophagy in aging and disease[END_REF]. On the other hand, the proteasome is inhibited by lipofuscin, perhaps as a result of binding proteasomal complexes in unsuccessful attempts of degradation. Moreover, it has been shown by different authors that the lipofuscinoccupied volume reaching maximal values of about 40% in the cytoplasm of aged animal cells and up to 75% in the motor neurons of human centenarians [START_REF] Szweda | Aging, lipofuscin formation, and free radical-mediated inhibition of cellular proteolytic systems[END_REF]. Comparing tissue samples from several organs (heart, liver, skeletal muscle, lung, kidney, cerebellum, and testis) of rats of different ages (2, 11, 29.5 months) showed a low amount of lipofuscin in the youngest animals and significant increases, up to 28-fold, in the oldest ones (29.5 months) [START_REF] Jung | Lipofuscin: formation, distribution, and metabolic consequences[END_REF].

It is important also remember that another marker linked to age is the increase, in Apo-E mice respect to controls, of polyploidy hepatocytes. This is in agreement with other results reported from Sanjeev [START_REF] Gupta | Hepatic polyploidy and liver growth control[END_REF] indicating that hepatic polyploidy is an advancement toward terminal differentiation and cellular senescence with greater probabilities of apoptosis.

Another important point of this study has been the evaluation of pro-inflammatory cytokines (IL-6 and iNOS). Their immunolocalization increases in E 0 mice with age, whereas it was very low in control animals and in young (6 week old) E° mice. There is accumulating evidence that aging is associated with a pro-inflammatory shift in vascular phenotype (Csizar et al., 2007), kidney [START_REF] Rodwell | A transcriptional profile of aging in the human kidney[END_REF], liver [START_REF] Coban | The effect of caffeic acid phenethyl ester (CAPE) against cholestatic liver injury in rats[END_REF] and heart [START_REF] Brinkley | Chronic inflammation is associated with low physical function in older adults across multiple comorbidities[END_REF]. We must comment that the presence of IL-6 in epithelial cells could be odd, but this result is in agreement with other data suggesting its expression in progressive renal disease [START_REF] Zhang | Role of NAD(P)H oxidase in transforming growth factor-1-induced monocyte chemoattractant protein-1 and interleukin-6 expression in rat renal tubular epithelial cells[END_REF]. In fact, these Authors proposed that ROS are likely involved in the expression of inflammatory cytokines such as MCP-1 and IL-6 production in renal tubular epithelial cells. Moreover, our results showing the increase of IL-6 expression in cardiomyocytes are consistent with the experiments carried out from Kurdi et al. (2009). Kurdi et al. reported that the mechanical stretch of cardiomyocytes presented an increased expression of IL-6. Increased production of ROS, normally due to a decrease of antioxidant enzymes (i.e. catalase, SOD), as we showed in all studied organs, is thought to be implicated in the inflammatory process by acting as second messenger; the pro-inflammatory cytokines are able to induce an increased degree of interstitial fibrosis and an influx of infiltrating mononuclear cells, suggesting that inflammation plays an important role in age-related pathological alterations in the kidney (Csizar et al., 2007). We showed increased fibrosis, not only in the kidney, but also in the liver and heart probably linked to the age and we propose that these organs show similar age-dependent metabolic changes. In particular, the age-dependent pro-inflammatory cytokines could induce an increased number of fibroblasts that produce extracellular matrix which inter alia contains collagen and elastin, important proteins for the structural integrity of the organs.

It is possible that with age, the relative amounts of the different proteins produced are changed; the ratio of newly formed to constitutive collagen is shifted towards the fibrotic one and there are also structural changes in elastin resulting in reduced elastic recoil. protein very important in the pathology of atherosclerosis, as previously reported, and also as a useful player in longevity and age-related diseases, i.e. inflammatory status that is a risk factor for mortality and functional decline. It is possible that the data presented may not be a direct result of apolipoprotein E deficiency, but rather that of apolipoprotein E induced oxidative stress, with changes in paraoxonases (PONs) and/or the development of atherosclerosis, though further studies would be needed to resolve this point. We consider that further studies of apoE protein could be applied in other areas, included its role, for example, in cancer, rheumatoid arthritis and macular degeneration.

Legends to figures

Figure 1 .

 1 Figure 1. Photomicrographs showing kidney of control (A) and 20 weeks old E° (B) mice. Sirius Red staining and polarization optics (A and B), Hematoxylin-eosin staining (insets). Bar: 10 m.

Figure 2 .

 2 Figure 2. Histograms showing the tubular number (A) and the percentage of glomerulosclerosis (B) in kidney of control and of E° mice 6 and 20 weeks old. *P<0.05 vs. Control. **P<0.05 vs. 6

Figure 3 .

 3 Figure 3. Photomicrographs showing IL-6 immunostaining in the kidney of control (A) and of E° mice 20 week old (B). Bar: 10 m.

Figure 4 .

 4 Figure 4. Histograms showing quantitative analysis (IOD) of iNOS (A) and endogenous antioxidant enzymes (B) in kidney, liver and heart of control and of E° mice 6 and 20 weeks old.

Figure 5 .

 5 Figure 5. Photomicrographs showing liver of control (A) and 20 weeks old E° (B) mice. PAS staining. Bar: 10 m. (C) Histogram showing the quantitative analysis of lipofuscin content.

Figure 6 .

 6 Figure 6. Histogram showing percentage of polyploid hepatocytes in liver of control and of E° mice 6 and 20 weeks old. *P<0.05 vs. Control. Hematoxylin-eosin staining.

Figure 7 .

 7 Figure 7. Photomicrographs showing IL-6 immunostaining in the liver of control (A) and of E° mice 20 weeks old (B). Bar: 10 m.

Figure 8 .

 8 Figure 8. Photomicrographs showing heart of control (A) and E° (B) mice 20 weeks old. Sirius Red staining with polarization optics (A and B), hematoxylin-eosin staining (insets). Bar: 10 m.

Figure 9 .

 9 Figure 9. Photomicrographs showing IL-6 immunostaining in the heart of control (A) and of E° mice (B) 20 weeks old. Bar: 10 m.

Acknowledgement:

We thank Dr R. Coates for the English revision of the manuscript. This study was supported by the grant (ex-60%) of the University of Brescia.


James A R Marshall
email: james.marshall@bristol.ac.uk

The Donation Game with Roles Played Between Relatives

Keywords: evolutionary game theory, 2 x 2 games, kin selection, donation

We consider a social game with two choices, played between two relatives, where roles are assigned to individuals so that the interaction is asymmetric. Behaviour in each of the two roles is determined by a separate genetic locus. Such asymmetric interactions between relatives, in which individuals occupy different behavioural contexts, may occur in nature, for example between adult parents and juvenile offspring. The social game considered is known to be equivalent to a donation game with non-additive payoffs, and has previously been analysed for the single locus case, both for discrete and continuous strategy traits. We present an inclusive fitness analysis of the discrete trait game with roles and recover equilibrium conditions including fixation of selfish or altruistic behaviour under both behavioural contexts, or fixation of selfish behaviour under one context and altruistic behaviour under the other context. These equilibrium solutions assume that the payoff matrices under each behavioural context are identical. The equilibria possible do depend crucially, however, on the deviation from payoff additivity that occurs when both interacting individuals act altruistically.

A c c e p t e d m a n u s c r i p t Introduction

Social behaviour is often considered in its simplest form as an interaction between two individuals, each of which can exhibit two alternative behaviours. It is assumed that the two individuals' choices are made independently, and that each receives a payoff from the game according to their own action and that of the other. Such a game can be written in its most general form as a payoff matrix of the following form (table 1). [START_REF] Grafen | The hawk-dove game played between relatives[END_REF] labelled the above payoff matrix the 'Hawk-Dove' game, not to be confused with the particular instantiation of the above payoff matrix labelled 'Hawk-Dove' by Maynard Smith (Maynard [START_REF] Smith | The logic of animal conflict[END_REF][START_REF] Smith | Evolution and the Theory of Games[END_REF]. By choosing different parameters A, B, C and D in this game a variety of social dilemmas can be realised [START_REF] Rapoport | A taxonomy of 2 x 2 games[END_REF][START_REF] Rapoport | Exploiter, leader, hero, and martyr: the four archetypes of the 2 x 2 game[END_REF], including the widely studied Prisoner's Dilemma and Snowdrift games. The study of such social dilemmas between related individuals has attracted much attention since the introduction of the concept of inclusive fitness (Hamilton, 1964a;1964b). Inclusive fitness theory considers altruistic interactions between individuals, where altruism is defined as incurring a fitness cost c to the altruist, while providing a fitness benefit b to the recipient. This is often referred to as the donation game. In fact, more general social interactions can be considered by allowing costs c or benefits b to be negative, such as weak altruism (c is negative, hence 'altruism' also increases personal fitness (Wilson, 1979) and even spite (b is negative, hence 'altruism' actually decreases the recipients' personal fitness [START_REF] Hamilton | Selfish and spiteful behaviour in an evolutionary model[END_REF]. In this paper, we limit ourselves to the simple donation game with positive c and b, corresponding to strong altruism (c is positive,

ibid.).

A c c e p t e d m a n u s c r i p t

If altruism is under genetic control, and if some population structure exists such that genetically similar individuals interact with each other more frequently than they interact with individuals drawn uniformly at random from the entire population (e.g. [START_REF] Grafen | The hawk-dove game played between relatives[END_REF], then altruistic behaviour is selected for whenever Hamilton's rule is satisfied (Hamilton, 1964a;1964b):

b c r > , ( 1
)
where r is the average degree of relatedness between interacting individuals.

Games having the form shown in table 1 played between relatives have been analysed

both for populations of fixed strategy individuals, and populations of individuals whose strategy is a probability of playing each of the two game alternatives. Different strategy equilibria have been demonstrated for each situation [START_REF] Grafen | The hawk-dove game played between relatives[END_REF].

By defining

D B c - = - , (2a)
D C b - = , (2b)
and

D C B A d + - - = , ( 2c
)
and by transforming the payoffs such that D = 0 1 , the payoff matrix in table 1 can be rewritten as table 2 below [START_REF] Queller | Kin selection and frequency dependence: a game theoretic approach[END_REF]).

1 game theoretic equilibria are invariant under such a payoff matrix transformation (e.g. Weibull, 1995, p. 73)

A c c e p t e d m a n u s c r i p t

Table 2 is the payoff matrix for the synergistic donation game in which each individual can decide whether to behave altruistically (choose I) or not (choose II).

Note that for strong altruism we require that b, c > 0, as described above. One important point is that there is a deviation from payoff additivity d, also known as the synergistic effect, that occurs when both individuals behave altruistically towards positively non-additive payoffs) or unstable (d < 0, i.e. negatively non-additive payoffs) results [START_REF] Queller | Kin selection and frequency dependence: a game theoretic approach[END_REF].

Games With Roles Played Between Relatives

Analyses of the 2 x 2 games, and the donation game in particular, often assume a symmetric interaction in which both players have opportunities to donate altruistically towards the other, and do so using a common decision rule. having the same outcome can occur when the payoff matrices are identical for each individual. In the remainder of this paper we will examine conditions for 'uncorrelated asymmetries' to occur in the donation game with roles played between relatives.

The Donation Game with Roles and Relatedness

For the model described below we will adopt the usual 'phenotypic gambit' [START_REF] Grafen | Natural selection, kin selection and group selection[END_REF], and assume that each behaviour is under independent genetic control by a haploid locus. We consider the game defined by the payoff matrix of table 1, which as noted above can be mapped onto the donation game with potentially non-additive payoffs as presented in table 2. We assume discrete, deterministic strategies. Thus far, our model is entirely the same as the discrete trait models considered by [START_REF] Grafen | The hawk-dove game played between relatives[END_REF] and [START_REF] Queller | Kin selection and frequency dependence: a game theoretic approach[END_REF]. We now make two modifications to these previous approaches: first, by considering two distinct behavioural contexts, with action under each being subject to a separate genetic locus. These two contexts are mutually behaviour in each context, and solve for fitness inequality. Subsequently we shall examine the relationship between selection pressure at these two behavioural loci using the replicator dynamics. Under the above model, these equations for a single behavioural context are

)) 1 ( ( ) 1 ( f C Af r f B Af w I - + + - + = , (3a) and )
) 1 ( ( ) 1 ( f D Bf r f D Cf w II - + + - + = , (3b)
where, as described above, r is relatedness for the locus in question, and f is the population frequency of altruistic behaviour at the locus for the other behavioural equations avoid double-counting the fitness effects of genes [START_REF] Grafen | How not to measure inclusive fitness[END_REF]1984), as fitness benefits of genes at two distinct loci are considered (cf. [START_REF] Queller | The measurement and meaning of inclusive fitness[END_REF].

There are two important quantities of interest to solve for in considering when (  )

D C D C B A f D B D C B A f r - + + - - - + + - - - = ′ . (4)
Through rewriting using equations 2a-2c this gives a synergistic version of Hamilton's rule

′ r = c -fd b + fd , ( 5

A c c e p t e d m a n u s c r i p t

If instead we solve w I = w II for the population frequency of altruistic behaviour in the other, opponent's, context above which altruistic behaviour will be favoured in the current behavioural context, then we have

′ f = D -B -r(C -D) (1+ r)(A -B -C + D) . ( 6
)
Intriguingly, this is identical to Grafen's equilibrium frequency for altruism in the continuous 'Hawk-Dove' game between relatives (Grafen, 1979, eq. 10), although here equation 6 above is actually the threshold frequency for altruistic behaviour under the opponent's behavioural context, above which altruistic behaviour at the focal individual's behavioural context will be selected for. Rewriting using equations 2a-2c, equation 6 can be expressed in terms of fitness costs, benefits and synergistic effects as

′ f = c -rb (1+ r)d . ( 7
)
It is easy to examine the fixed points of selection at the loci determining behaviours in both roles by substituting equations 3a and 3b into the replicator dynamics [START_REF] Schuster | Replicator dynamics[END_REF]), so we consider change in frequency of altruistic behaviour for the focal individual's behavioural context, g, as

( ) ( ) (
)

II I I w g gw w g g - + - = 1 & . ( 8

A c c e p t e d m a n u s c r i p t

The equilibria of equation 8 are, of course, where g = 0, g = 1, and w I = w II . In all cases this last equilibrium will be unstable. Selection of the remaining two equilibria according to the replicator dynamics will depend on relative costs c and benefits b of altruism, the relatedness r in the population, and the synergistic effect d. figure 1). If, however, the synergistic effect is non-zero, and if population relatedness is at an intermediate value, such that it falls between the extrema of equation 5, then more interesting dynamics are observed. If the synergistic effect d is negative, then selective pressures act to bring about a stable co-existence of an altruistic behaviour for one behavioural context and a selfish behaviour for the other behavioural context (figure 2). Which of these equilibria is selected will depend on the population frequencies of the two behaviours and the relative strength of selection at the two loci.

Conversely, if the synergistic effect d is positive, then either fixation of altruistic behaviour under both contexts, or selfish behaviour under both contexts, is stable (figure 3). Again, which equilibrium is selected depends on population frequencies and relative strengths of selection for the two behaviours. The stability of the different equilibria is demonstrated in the Appendix A. These results agree with a derivation based on Maynard [START_REF] Smith | Evolution and the Theory of Games[END_REF] work, as demonstrated in Appendix B.

Discussion

A c c e p t e d m a n u s c r i p t

Evolutionary game theory has been applied to the study of altruistic interactions between relatives where there may be a synergistic deviation from payoff additivity when two altruists interact [START_REF] Grafen | The hawk-dove game played between relatives[END_REF][START_REF] Queller | Kin selection and frequency dependence: a game theoretic approach[END_REF]. In earlier models a single altruistic trait has been considered. In this paper we have examined the synergistic donation game played between relatives when each individual occupies a different behavioural context, and different altruistic traits govern behaviour under each behavioural context. In such games stable equilibria corresponding to fixation of altruistic behaviour under both contexts, or fixation of selfish behaviour under both contexts, are both possible according to payoffs and relatedness. Negatively nonadditive payoffs and intermediate levels of relatedness however, induce a change in the stable equilibria of the population: negatively non-additive payoffs and a suitably intermediate relatedness result in stable pairings of altruism under one behavioural context, and selfishness under the other behavioural context. In contrast, positively non-additive payoffs with a suitably intermediate relatedness result in stable fixation of altruistic behaviour under both contexts, or stable fixation of selfish behaviour under both contexts. These results for the donation game with roles are analogous to results for the single trait case in which negatively non-additive payoffs induce stable polymorphic population equilibria, and positively non-additive payoffs induce unstable polymorphic population equilibria [START_REF] Queller | Kin selection and frequency dependence: a game theoretic approach[END_REF].

Situations in which individuals involved in an altruistic interaction do so under different behavioural contexts are likely to occur frequently in nature. For example, the use of state-dependent behavioural rules leads to such asymmetric interactions (e.g. Maynard [START_REF] Smith | The logic of asymmetric contests[END_REF][START_REF] Selten | A note on evolutionary stable strategies in asymmetric animal conflicts[END_REF][START_REF] Selten | Evolutionary stability in extensive 2-person games[END_REF][START_REF] Hammerstein | The role of asymmetries in animal contests[END_REF][START_REF] Smith | Evolution and the Theory of Games[END_REF][START_REF] Houston | Models of Adaptive Behaviour: An Approach Based on State[END_REF]). We consider cases in which

  each other. The game payoffs A, B, C and D can be chosen such that d = 0 (payoffs are additive), d < 0 (payoffs are negatively non-additive), or d > 0 (payoffs are positively non-additive), while still maintaining b, c > 0. Deviation from payoffadditivity has implications for the application of Hamilton's rule. For the synergistic donation game played in a population of relatives with deterministic strategies, the population strategy at equilibrium depends both on relatedness r and on deviation from additivity of payoffs d. If relatedness is sufficiently high or low, or if payoffs are additive (d = 0) altruistic (type I) or selfish (type II) behaviour will take over the population. However if relatedness is intermediate and payoffs are non-additive (d ≠ 0) then a polymorphic population equilbrium which is either stable (d > 0, i.e.

A c c e p t e d m a n u s c r i p t

  exclusive for any given interaction, so when two players interact each must occupy a different behavioural context, and thus use different loci to determine their behaviour for the interaction. We consider a population in which individuals in one role have type I (hereafter called altruistic) behaviour at population frequency g, and type II (hereafter called selfish) behaviour at population frequency (1g). We model the inclusive fitness of altruistic and selfish behaviour to determine the change in g over time, given that these individuals encounter altruistic behaviour in opponents occupying the other role at the population frequency f, and selfish behaviour at population frequency (1f). A similar calculation is done for the change in f over time, given g. Second, to model the inclusive fitness component of the interaction, the payoff the opponent receives from the interaction is included in the fitness calculation, weighted by the degree of relatedness r for interactions within the population. We shall not consider the frequency with which individuals in the population experience each behavioural context. Rather, we shall write down inclusive fitness equations for the alleles for altruistic (type I) and selfish (type II)

A c c e p t e d m a n u s c r i p t

  context. In equations 3a and 3b above the first two terms give the expected payoff to the focal individual, and the last term is the expected payoff to the opponent, weighted by the focal individual's relatedness to them. The above inclusive fitness

  altruistic and selfish behaviours have equal fitness. If we solve w I = w II for the critical relatedness threshold r′ above which altruistic behaviour is favoured by selection we have ( )

  ) which is the equivalent for our discrete donation game with roles of[START_REF] Queller | Kin selection and frequency dependence: a game theoretic approach[END_REF] equation 5b for the discrete donation game without roles. If payoffs are additive (d = 0) then equation 5 reduces to Hamilton's rule, however as discussed in Queller and above positively non-additive payoffs (d > 0) make altruism easier to select for than predicted by Hamilton's rule, while for negatively non-additive payoffs (d < 0) altruism is harder to select than Hamilton's rule predicts. In both cases the relatedness threshold for altruism varies non-linearly as a function of frequency of altruism in the population under the other behavioural context.

) Equations 3

 3 to 8 all refer to selection operating on one of the behavioural contexts. By symmetry, equations for the other behavioural context are obtained by simply exchanging f for g and vice-versa throughout. This gives an equation for Ý f , the change in f over time, that is the counterpart of equation 8, allowing us simultaneously to consider the change in g and f over time.

Figures 1 to 3

 3 show numerically calculated sample dynamics of equation 8 (and its counterpart for Ý f ), for representative values of these parameters. If ratio of cost to benefit is sufficiently high, or low, relative to population relatedness, then at both behavioural contexts alike altruistic or selfish behaviour will go to fixation regardless of population state, just as in the standard donation game with additive payoffs (e.g.

A c c e p t e d m a n u s c r i p tA c c e p t e d m a n u s c r i p tA c c e p t e d m a n u s c r i p t 21 .

 21 payoffs and actions are the same from both players' points of view, but each player makes use of a different trait to determine their action. One particular example of when such situations can occur is during repeated interaction with memory, where both individuals base their decisions on the outcome of their previous interaction. If at one interaction one individual provides an altruistic benefit to the other, but receives nothing in return, then at the next interaction each individual has a different perspective: one individual has just exploited the other, while the other has just been exploited. Marshall & Rowe (2003) studied such games and concluded, based on the traditional Prisoner's Dilemma payoff matrix, that interactions between relatives could undermine selection for 'retaliatory' behaviour in the Iterated Prisoner's Dilemma, in which an individual responds to exploitation of the form just described above by behaving selfishly on the next interaction. Marshall & Rowe concluded that kin selection could thus undermine selection for conditionally cooperative strategies for repeated interactions, such as the famous Tit-for-Tat[START_REF] Axelrod | The Evolution of Cooperation[END_REF]. The results presented here show that this effect is in fact due to the negatively non-additive nature of the traditional Prisoner's Dilemma payoffs, and that a variety of other outcomes could occur given additive or positively non-additive game payoffs.Games played between relatives using different behavioural rules as they occupy different behavioural contexts are likely occur in many other situations in nature.[START_REF] Queller | The measurement and meaning of inclusive fitness[END_REF] discussed conditional expression of traits based on state variables, such as in the example of a stronger individual monopolising the role of reproductive and relegating the weaker relative to the role of helper, and sketched an outline of our model by considering interactions in which individuals express different genes to make their game choices. We agree with Queller's observation that conditional expression of traits is likely to be more common than unconditional expression due to the greater flexibility of the former strategy (see also[START_REF] Houston | Models of Adaptive Behaviour: An Approach Based on State[END_REF].In cases such as the example of a stronger individual monopolising reproduction, differences in the resources of the individuals might well be expected to lead to different payoff matrices for the interactants. In such cases it is of course easy to envisage payoff matrices that induce one individual to behave selfishly and the other to behave altruistically, due to the different costs and benefits of altruism that would accrue to each. However Maynard[START_REF] Smith | The logic of asymmetric contests[END_REF] have pointed out that differences in payoffs are not necessary for such equilibria to arise, and that 'uncorrelated asymmetries' of the same form can arise even when the payoff matrices are the same for both interacting individuals. The scenario presented in Marshall & Rowe (2003) and outlined above is an example of just such an uncorrelated asymmetry. The contribution of this paper is an explicit analysis of how uncorrelated asymmetries can arise in donation games with roles played between relatives, in terms of a modified version of Hamilton's rule. This analysis highlights the crucial importance of negative non-additivity of payoffs in such interactions. Given it seems likely that non-additive interactions occur frequently in nature, it is hoped that the results presented here will be of some value. Wilson, D. S., 1979. Structured demes and trait-group variation. Am. Nat. 113, 606-610.

Figure 1 :Figure 2 :Figure 3 : 1 .

 1231 Figure 1: Selection pressure on frequency of altruistic behaviour under first context (f) and second context (g) for additive payoffs and high relatedness (b = 4, c = 2, d = 0, r = 0.6)

Table 2 .

 2 Payoffs for the donation game with potentially non-additive payoffs (payoffs to row player shown).

  TableB1. Payoffs for the 'uncorrelated asymmetry' analysis (payoffs to row player shown).

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Fig 1 :

 1 Fig 1:

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Fig 2 :

 2 Fig 2:

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Fig 3 :

 3 Fig 3:

A c c e p t e d m a n u s c r i p t interactions

  between relatives. For example, in a potentially altruistic interaction between an adult and offspring each individual may occupy different behavioural contexts, and hence use different rules to decide on altruistic behaviour, through differences in individuals' state variables such as energy reserves and body mass, possession of a territory, and so on. In such situations it may provide an inclusive fitness benefit to an individual to behave altruistically in one context, but not in another. Maynard[START_REF] Smith | The logic of asymmetric contests[END_REF] have also noted that one need not assume

	However in nature many
	typical social interactions are not symmetric, but individuals may interact under
	different behavioural contexts, making use of different rules to determine behaviour
	in each context (e.g. Maynard Smith & Parker, 1976; Selten, 1980; 1983,

differences in payoffs under the different contexts for such equilibria, as one would expect if individuals' states did indeed differ, but that 'uncorrelated asymmetries'

Acknowledgements

I thank A. I. Houston and two anonymous reviewers for helpful comments and suggestions.

Appendix A

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We consider the stability of the various equilibria under the replicator dynamics (equation 7), which for ease of exposition can be written as

making clear that selection always acts in favour of the fittest allele, since as g ∈ [0,1] then g -g 2 ≥ 0. Now it is easy to see that the w Iw II equilibrium is unstable since, as w Iw II does not depend on g and there can thus be no frequency dependent effects in g, an increase in the fitness of one allele will lead to runaway positive selection for that allele.

For the stability of the f ∈ {0,1}, g ∈ {0,1} equilibria we first choose f = 0. Then

Next we choose f = 1. Then

which is negative when r is less than the r´ given by equation 5 with

By symmetry the same analysis can be applied (exchanging f for g and vice-versa throughout) to find the stability of the f = 0 and f = 1 equilibria.

Putting these conditions together, the f = 0, g = 1 and f = 1, g = 0 equilibria are bistable iff

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Conversely both the f = 0, g = 0 and f = 1, g = 1 equilibria are bistable iff

Now assuming that we are considering strong altruism, so b and c are both positive, then

iff d < 0. So the f = 0, g = 1 and f = 1, g = 0 equilibria are bistable iff inequality A4 is satisfied, which can only occur if d < 0 (payoffs are negatively non-additive), whereas the f = 0, g = 0 and f = 1, g = 1 equilibria are bistable iff inequality A5 is satisfied, which can only occur if d > 0 (payoffs are positively non-additive). Otherwise one of the f = 0, g = 0 or f = 1, g = 1 equilibria will be globally stable.

Appendix B

We show the relationship between the derivation of the results presented in this paper, and a derivation based on Maynard Smith's (1982, pp.123-126) 'uncorrelated asymmetries' approach.

We first assign values to Maynard Smith's payoff matrix (table B1) to reflect the inclusive fitness of type I and type II behaviour given the opponent's behaviour, based on equations 3a and 3b. If the opponent displays type I behaviour then f = 1 in these equations, otherwise f = 0. Thus the payoffs in the game are α = (1+ r)A , (B1a) If inequalities B3a and B3b are satisfied then one can rewrite them using equations B1a-B1d and 2a-c and combine to give inequality A4, whereas if neither inequality is satisfied then we can rewrite as inequality A5. Thus the results of appendix A can also be derived using Maynard Smith's results.