ABSTRACT
OBJECTIVE: In order to facilitate data pooling between studies, we explored harmonisation of patient-reported outcome measures (PROMs) in people with knee pain due to osteoarthritis or knee trauma, using the Patient Acceptable Symptom State scores (PASS) as a criterion. METHODS: We undertook a systematic literature review (SLR) of PASS scores, and performed individual participant data (IPD) analysis of score distributions from concurrently completed PROM pairs. Numerical rating scales (NRS), visual analogue scales, KOOS and WOMAC pain questionnaires were standardised to 0 to 100 (worst) scales. Meta-regression explored associations of PASS. Bland Altman plots compared PROM scores within individuals using IPD from WebEx, KICK, MenTOR and NEKO studies. RESULTS: SLR identified 18 studies reporting PASS in people with knee pain. Pooled standardised PASS was 27 (95% CI: 21 to 35; n = 6,339). PASS was statistically similar for each standardised PROM. Lower PASS was associated with lower baseline pain (ß = 0.49, P = 0.01) and longer time from treatment initiation (Q = 6.35, P = 0.04). PASS scores were lowest in ligament rupture (12, 95% CI: 11 to 13), but similar between knee osteoarthritis (31, 95% CI: 26 to 36) and meniscal tear (27, 95% CI: 20 to 35). In IPD, standardised PROMs each revealed similar group mean scores, but scores within individuals diverged between PROMs (LoA between -7 to -38 and +25 to 52). CONCLUSION: Different standardised PROMs give similar PASS thresholds in group data. PASS thresholds may be affected more by patient and treatment characteristics than between PROMs. However, different PROMs give divergent scores within individuals, possibly reflecting different experiences of pain.
Subject(s)
Knee Injuries , Osteoarthritis, Knee , Humans , Patient Reported Outcome Measures , Knee Joint , Osteoarthritis, Knee/complications , Osteoarthritis, Knee/therapy , PainABSTRACT
OBJECTIVE: MicroRNA 140 (miR-140) is a chondrocyte-specific endogenous gene regulator implicated in osteoarthritis (OA). As mechanical injury is a primary aetiological factor in OA, we investigated miR-140-dependent mechanosensitive gene regulation using a novel CRISPR-Cas9 methodology in primary human chondrocytes. METHOD: Primary (passage 1/2) human OA chondrocytes were isolated from arthroplasty samples (six donors) and transfected with ribonuclear protein complexes or plasmids using single guide RNAs (sgRNAs) targeting miR-140, in combination with Cas9 endonuclease. Combinations of sgRNAs and single/double transfections were tested. Gene editing was measured by T7 endonuclease 1 (T7E1) assay. miRNA levels were confirmed by qPCR in chondrocytes and in wild type murine femoral head cartilage after acute injury. Predicted close match off-targets were examined. Mechanosensitive miR-140 target validation was assessed in 42 injury-associated genes using TaqMan Microfluidic cards in targeted and donor-matched control chondrocytes. Identified targets were examined in RNAseq data from costal chondrocytes from miR-140-/- mice. RESULTS: High efficiency gene editing of miR-140 (90-98%) was obtained when two sgRNAs were combined with double RNP-mediated CRISPR-Cas9 transfection. miR-140 levels fell rapidly after femoral cartilage injury. Of the top eight miR-140 gene targets identified (P < 0.01), we validated three previously identified ones (septin 2, bone morphogenetic protein 2 and fibroblast growth factor 2). Novel targets included Agrin, a newly recognised pro-regenerative cartilage agent, and proteins associated with retinoic acid signalling and the primary cilium. CONCLUSION: We describe a highly efficient CRISPR-Cas9-mediated strategy for gene editing in primary human chondrocytes and identify several novel mechanosensitive miR-140 targets of disease relevance.
Subject(s)
MicroRNAs , Osteoarthritis , Animals , CRISPR-Cas Systems , Chondrocytes/metabolism , Humans , Joints/metabolism , Mice , MicroRNAs/metabolism , Osteoarthritis/metabolismABSTRACT
Preclinical imaging in osteoarthritis is a rapidly growing area with three principal objectives: to provide rapid, sensitive tools to monitor the course of experimental OA longitudinally; to describe the temporal relationship between tissue-specific pathologies over the course of disease; and to use molecular probes to measure disease activity in vivo. Research in this area can be broadly divided into those techniques that monitor structural changes in tissues (microCT, microMRI, ultrasound) and those that detect molecular disease activity (positron emission tomography (PET), optical and optoacoustic imaging). The former techniques have largely evolved from experience in human joint imaging and have been refined for small animal use. Some of the latter tools, such as optical imaging, have been developed in preclinical models and may have translational benefit in the future for patient stratification and for monitoring disease progression and response to treatment. In this narrative review we describe these methodologies and discuss the benefits to animal research, understanding OA pathogenesis, and in the development of human biomarkers.
Subject(s)
Joints/diagnostic imaging , Osteoarthritis/diagnostic imaging , Animals , Bone and Bones/diagnostic imaging , Cartilage, Articular/diagnostic imaging , Disease Models, Animal , Magnetic Resonance Imaging , Mice , Molecular Imaging , Optical Imaging , Osteophyte/diagnostic imaging , Photoacoustic Techniques , Positron-Emission Tomography , Synovial Membrane/diagnostic imaging , Ultrasonography , X-Ray MicrotomographyABSTRACT
OBJECTIVE: Surgical knee joint distraction (KJD) leads to clinical improvement in knee osteoarthritis (OA) and also apparent cartilage regeneration by magnetic resonance imaging. We investigated if alteration of the joint's mechanical environment during the 6 week period of KJD was associated with a molecular response in synovial fluid, and if any change was associated with clinical response. METHOD: 20 individuals undergoing KJD for symptomatic radiographic knee OA had SF sampled at baseline, midpoint and endpoint of distraction (6 weeks). SF supernatants were measured by immunoassay for 10 predefined mechanosensitive molecules identified in our previous pre-clinical studies. The composite Knee injury and OA Outcome Score-4 (KOOS4) was collected at baseline, 3, 6 and 12 months. RESULTS: 13/20 (65%) were male with mean age 54°±°5yrs. All had Kellgren-Lawrence grade ≥2 knee OA. 6/10 analytes showed statistically significant change in SF over the 6 weeks distraction (activin A; TGFß-1; MCP-1; IL-6; FGF-2; LTBP2), P < 0.05. Of these, all but activin A increased. Those achieving the minimum clinically important difference of 10 points for KOOS4 over 6 months showed greater increases in FGF-2 and TGFß-1 than non-responders. An increase in IL-8 during the 6 weeks of KJD was associated with significantly greater improvement in KOOS4 over 12 months. CONCLUSION: Detectable, significant molecular changes are observed in SF following KJD, that are remarkably consistent between individuals. Preliminary findings appear to suggest that increases in some molecules are associated with clinically meaningful responses. Joint distraction may provide a potential opportunity in the future to define regenerative biomarker(s) and identify pathways that drive intrinsic cartilage repair.
Subject(s)
External Fixators , Orthopedic Procedures/methods , Osteoarthritis, Knee/metabolism , Osteoarthritis, Knee/surgery , Synovial Fluid/metabolism , Activins/metabolism , Cell Adhesion Molecules/metabolism , Chemokine CCL2/metabolism , Female , Fibroblast Growth Factor 2/metabolism , Humans , Interleukin-6/metabolism , Interleukin-8/metabolism , Latent TGF-beta Binding Proteins/metabolism , Male , Matrix Metalloproteinase 3/metabolism , Middle Aged , Tissue Inhibitor of Metalloproteinase-1/metabolism , Transforming Growth Factor beta1/metabolism , Treatment OutcomeABSTRACT
OBJECTIVE: The role of inflammation in structural and symptomatic osteoarthritis (OA) remains unclear. One key mediator of inflammation is the chemokine CCL2, primarily responsible for attracting monocytes to sites of injury. We investigated the role of CCL2 and its receptor CCR2 in experimental OA. DESIGN: OA was induced in 10 weeks old male wild type (WT), Ccl2-/- and Ccr2-/- mice, by destabilisation of the medial meniscus (DMM). RNA was extracted from whole joints at 6 h and 7 days post-surgery and examined by reverse transcription polymerase chain reaction (RT-PCR). Gene expression changes between naïve and DMM-operated mice were compared. Chondropathy scores, from mice at 8, 12, 16 and 20 weeks post DMM were calculated using modified Osteoarthritis Research Society International (OARSI) grading systems. Changes in hind paw weight distribution, as a measure of pain, were assessed by Linton incapacitance. RESULTS: Absence of CCL2 strongly suppressed (>90%) selective inflammatory response genes in the joint 6 h post DMM, including arginase 1, prostaglandin synthase 2, nitric oxide synthase 2 and inhibin A. IL6, MMP3 and tissue inhibitor of metalloproteinase 1 were also significantly suppressed. Similar trends were also observed in the absence of CCR2. A lower average chondropathy score was observed in both Ccl2-/- and Ccr2-/- mice at 12, 16 and 20 weeks post DMM compared with WT mice, but this was only statistically significant at 20 weeks in Ccr2-/- mice. Pain-related behaviour in Ccl2-/- and Ccr2-/- mice post DMM was delayed in onset. CONCLUSION: The CCL2/CCR2 axis plays an important role in the development of pain in murine OA, but contributes little to cartilage damage.
Subject(s)
Chemokine CCL2/metabolism , Osteoarthritis/metabolism , Pain/metabolism , Receptors, CCR2/metabolism , Animals , Disease Models, Animal , Gene Expression , Male , Mice , Mice, Inbred C57BL , Mice, Knockout , Osteoarthritis/etiology , Osteoarthritis/pathology , Pain/etiology , Pain/pathology , Reverse Transcriptase Polymerase Chain ReactionABSTRACT
Osteoarthritis is a major source of pain, disability, and socioeconomic cost worldwide. The epidemiology of the disorder is complex and multifactorial, with genetic, biological, and biomechanical components. Aetiological factors are also joint specific. Joint replacement is an effective treatment for symptomatic end-stage disease, although functional outcomes can be poor and the lifespan of prostheses is limited. Consequently, the focus is shifting to disease prevention and the treatment of early osteoarthritis. This task is challenging since conventional imaging techniques can detect only quite advanced disease and the relation between pain and structural degeneration is not close. Nevertheless, advances in both imaging and biochemical markers offer potential for diagnosis and as outcome measures for new treatments. Joint-preserving interventions under development include lifestyle modification and pharmaceutical and surgical modalities. Some show potential, but at present few have proven ability to arrest or delay disease progression.
Subject(s)
Osteoarthritis/diagnosis , Osteoarthritis/therapy , Biomarkers/metabolism , Disease Progression , Early Diagnosis , Humans , Life Style , Osteoarthritis/epidemiologyABSTRACT
Age is the strongest independent risk factor for the development of osteoarthritis (OA) and for many years this was assumed to be due to repetitive microtrauma of the joint surface over time, the so-called 'wear and tear' arthritis. As our understanding of OA pathogenesis has become more refined, it has changed our appreciation of the role of ageing on disease. Cartilage breakdown in disease is not a passive process but one involving induction and activation of specific matrix-degrading enzymes; chondrocytes are exquisitely sensitive to changes in the mechanical, inflammatory and metabolic environment of the joint; cartilage is continuously adapting to these changes by altering its matrix. Ageing influences all of these processes. In this review, we will discuss how ageing affects tissue structure, joint use and the cellular metabolism. We describe what is known about pathways implicated in ageing in other model systems and discuss the potential value of targeting these pathways in OA.
Subject(s)
Aging/genetics , Cellular Senescence/genetics , Osteoarthritis/genetics , Aging/pathology , Cartilage/metabolism , Cartilage/physiopathology , Cellular Senescence/physiology , Chondrocytes/pathology , Chondrocytes/physiology , Humans , Osteoarthritis/physiopathology , Signal Transduction/geneticsABSTRACT
OBJECTIVE: Identify gene changes in articular cartilage of the medial tibial plateau (MTP) at 2, 4 and 8 weeks after destabilisation of the medial meniscus (DMM) in mice. Compare our data with previously published datasets to ascertain dysregulated pathways and genes in osteoarthritis (OA). DESIGN: RNA was extracted from the ipsilateral and contralateral MTP cartilage, amplified, labelled and hybridized on Illumina WGv2 microarrays. Results were confirmed by real-time polymerase chain reaction (PCR) for selected genes. RESULTS: Transcriptional analysis and network reconstruction revealed changes in extracellular matrix and cytoskeletal genes induced by DMM. TGFß signalling pathway and complement and coagulation cascade genes were regulated at 2 weeks. Fibronectin (Fn1) is a hub in a reconstructed network at 2 weeks. Regulated genes decrease over time. By 8 weeks fibromodulin (Fmod) and tenascin N (Tnn) are the only dysregulated genes present in the DMM operated knees. Comparison with human and rodent published gene sets identified genes overlapping between our array and eight other studies. CONCLUSIONS: Cartilage contributes a minute percentage to the RNA extracted from the whole joint (<0.2%), yet is sensitive to changes in gene expression post-DMM. The post-DMM transcriptional reprogramming wanes over time dissipating by 8 weeks. Common pathways between published gene sets include focal adhesion, regulation of actin cytoskeleton and TGFß. Common genes include Jagged 1 (Jag1), Tetraspanin 2 (Tspan2), neuroblastoma, suppression of tumourigenicity 1 (Nbl1) and N-myc downstream regulated gene 2 (Ndrg2). The concomitant genes and pathways we identify may warrant further investigation as biomarkers or modulators of OA.
Subject(s)
Cartilage, Articular/metabolism , Menisci, Tibial/metabolism , Microarray Analysis/methods , Osteoarthritis, Knee/etiology , Osteoarthritis, Knee/metabolism , Signal Transduction/physiology , Transcription, Genetic/physiology , Wounds and Injuries/complications , Adaptor Proteins, Signal Transducing , Animals , Calcium-Binding Proteins/metabolism , Cartilage, Articular/pathology , Cell Cycle Proteins , Disease Models, Animal , Extracellular Matrix Proteins/metabolism , Fibromodulin , Fibronectins/metabolism , Intercellular Signaling Peptides and Proteins/metabolism , Jagged-1 Protein , Male , Membrane Proteins/metabolism , Menisci, Tibial/pathology , Mice , Mice, Inbred C57BL , Nerve Tissue Proteins/metabolism , Osteoarthritis, Knee/pathology , Proteins/metabolism , Proteoglycans/metabolism , Serrate-Jagged Proteins , Signal Transduction/genetics , Tenascin/metabolism , Tetraspanins/metabolism , Transforming Growth Factor beta/metabolismABSTRACT
OBJECTIVE: Mouse articular cartilage (AC) is mostly assessed by histopathology and its mechanics is poorly characterised. In this study: (1) we developed non-destructive imaging for quantitative assessment of AC morphology and (2) evaluated the mechanical implications of AC structural changes. METHODS: Knee joints obtained from naïve mice and from mice with osteoarthritis (OA) induced by destabilization of medial meniscus (DMM) for 4 and 12 weeks, were imaged by phosphotungstic acid (PTA) contrast enhanced micro-computed tomography (PTA-CT) and scored by conventional histopathology. Our software (Matlab) automatically segmented tibial AC, drew two regions centred on each tibial condyle and evaluated the volumes included. A finite element (FE) model of the whole mouse joint was implemented to evaluate AC mechanics. RESULTS: Our method achieved rapid, automated analysis of mouse AC (structural parameters in <10 h from knee dissection) and was able to localise AC loss in the central region of the medial tibial condyle. AC thickness decreased by 15% at 4 weeks and 25% at 12 weeks post DMM surgery, whereas histopathology scores were significantly increased only at 12 weeks. FE simulations estimated that AC thinning at early-stages in the DMM model (4 weeks) increases contact pressures (+39%) and Tresca stresses (+43%) in AC. CONCLUSION: PTA-CT imaging is a fast and simple method to assess OA in murine models. Once applied more extensively to confirm its robustness, our approach will be useful for rapidly phenotyping genetically modified mice used for OA research and to improve the current understanding of mouse cartilage mechanics.
Subject(s)
Cartilage, Articular/diagnostic imaging , Knee Joint/diagnostic imaging , Osteoarthritis, Knee/diagnostic imaging , Tibia/diagnostic imaging , Animals , Biomechanical Phenomena , Cartilage, Articular/pathology , Cartilage, Articular/physiopathology , Contrast Media , Disease Models, Animal , Early Diagnosis , Imaging, Three-Dimensional , Knee Joint/pathology , Knee Joint/physiopathology , Menisci, Tibial/surgery , Mice , Osteoarthritis, Knee/pathology , Osteoarthritis, Knee/physiopathology , Phosphotungstic Acid , Tibia/pathology , Tibia/physiopathology , X-Ray MicrotomographyABSTRACT
OBJECTIVE: Alterations in joint mechanics can cause osteoarthritis, which results in degeneration of both cartilage and bone tissue. The objective of this work is to measure changes in the laxity of the mouse knee joint after destabilisation of the medial meniscus (DMM) and to visualise and quantify the resulting three-dimensional changes in the bone and cartilage. METHODS: Skeletally mature C57Bl6 male mice underwent DMM surgery in the right leg. Animals were sacrificed immediately 0 weeks (n=15), 4 weeks (n=11) or 8 weeks (n=12) after surgery. For the 0-week group, the anterior-posterior (AP) and varus-valgus laxity of the DMM limb were compared to the contralateral limb. For 4 and 8-week groups, tibiae were scanned with micro-computed tomography (µCT) to quantify and visualise bone changes and with confocal scanning laser microscopy (CSLM) to measure changes in cartilage. RESULTS: Laxity testing measured an increase in AP range of motion, particularly in the anterior direction. The DMM limbs showed a decrease in epiphyseal trabecular bone at 8 weeks and a decrease in cartilage volume, primarily on the posterior medial plateau, compared to the contralateral limb. Significant bone remodelling was observed at the periphery of the joint and in severe cases, osteolysis extended through the growth plate. CONCLUSION: Multimodal imaging allowed quantifiable 3D assessment of bone and cartilage and indicated extensive changes in the tissues. The increase in AP laxity suggests that DMM surgery redistributes loading posteriorly on the medial plateau, resulting in bone and cartilage loss primarily on the posterior portion of the medial plateau.
Subject(s)
Bone and Bones/pathology , Cartilage, Articular/pathology , Menisci, Tibial/surgery , Osteoarthritis/physiopathology , Analysis of Variance , Animals , Bone and Bones/diagnostic imaging , Cartilage, Articular/diagnostic imaging , Disease Models, Animal , Imaging, Three-Dimensional , Male , Menisci, Tibial/pathology , Mice , Mice, Inbred BALB C , Radiography , Range of Motion, Articular/physiologyABSTRACT
Joint surface defects (JSD) involving the articular cartilage and the subchondral bone are a common clinical problem in rheumatology and orthopaedics. The recent availability of accurate imaging for diagnosis and efficacious therapeutic options has stirred new interest in their natural history and biology. The evidence that some of these lesions can heal spontaneously whereas others precipitate osteoarthritis has raised important questions as to which lesions should be treated, when, and how. Evidence of repair of some of these lesions has also stimulated research into which factors contribute to successful healing and which ones determine chronic evolution and development of osteoarthritis (OA). Older anatomical observations, together with novel molecular tools and experimental models, have revealed a complex cellular and molecular response of cartilage to focal defects, which could explain differences in healing responses between individuals, and may provide clues to stimulating intrinsic tissue repair. In the first part of this review we will discuss clinical aspects of these lesions in the patient, with particular emphasis on their biology and natural history. In the second part we will summarize the data coming from in vitro and in vivo models of cartilage injury and regeneration, focussing on the molecular control of cartilage homeostasis after creation of cartilage surface defects.
Subject(s)
Cartilage, Articular/pathology , Animals , Cartilage Diseases/etiology , Cartilage Diseases/physiopathology , Humans , Osteoarthritis/pathologyABSTRACT
OBJECTIVE: The adamalysin metalloproteinase 15 (ADAM15) has been shown to protect against development of osteoarthritis in mice. Here, we have investigated factors that control ADAM15 levels in cartilage. DESIGN: Secretomes from wild-type and Adam15 -/- chondrocytes were compared by label-free quantitative mass spectrometry. mRNA was isolated from murine knee joints, either with or without surgical induction of osteoarthritis on male C57BL/6 mice, and the expression of Adam15 and other related genes quantified by RT-qPCR. ADAM15 in human normal and osteoarthritic cartilage was investigated similarly and by fluorescent immunohistochemistry. Cultured HTB94 chondrosarcoma cells were treated with various anabolic and catabolic stimuli, and ADAM15 mRNA and protein levels evaluated. RESULTS: There were no significant differences in the secretomes of chondrocytes from WT and Adam15 -/- cartilage. Expression of ADAM15 was not altered in either human or murine osteoarthritic cartilage relative to disease-free controls. However, expression of ADAM15 was markedly reduced upon aging in both species, to the extent that expression in joints of 18-month-old mice was 45-fold lower than in that 4.5-month-old animals. IL-13 increased expression of ADAM15 in HTB94 âcells by 2.5-fold, while modulators of senescence and autophagy pathways had no effect. Expression of Il13 in the joint was reduced with aging, suggesting this cytokine may control ADAM15 levels in the joint. CONCLUSION: Expression of the chondroprotective metalloproteinase ADAM15 is reduced in aging human and murine joints, possibly due to a concomitant reduction in IL-13 expression. We thus propose IL-13 as a novel factor contributing to increased osteoarthritis risk upon aging.
ABSTRACT
Human carcinogenesis is a multistep process in which epithelial cells progress through a series of premalignant phenotypes until an invasive cancer emerges. Extensive experimental observations in carcinogenesis have demonstrated this process can be divided into three general eras: initiation, promotion, and progression. However, this empirically derived, tissue-level explanation of carcinogenesis has not been reconciled with the step-wise genotypic and phenotypic changes encompassed in evolutionary paradigms such as the Feoron-Vogelstein diagram. Here, we analyze an evolutionary model of cellular dynamics that defines mutual interactions of cellular and subcellular events and tissue level changes in tumor growth and morphology. Results are expressed using an adaptive landscape that illustrates the evolutionary potential of cells that allow them to adapt to specific microenvironmental selection forces. It is shown that normal epithelial cells have a novel adaptive landscape that permits coexistence of normal cellular populations but also allows invasion by mutant phenotypes. Subsequent cancer evolution is possible due to a relaxation of tissue growth constraints (as mediated by cell-cell and cell-extracellular matrix interactions) and adaptations in response to perturbations in microenvironmental substrate concentrations (due to separation of evolving tumor cells from their blood supply by an intact basement membrane). Simulations, based on the dynamic model, produce three distinct stages of carcinogenesis that are consistent with the initiation, promotion, and progression stages observed experimentally. The simulations provide insight into the underlying cellular and microenvironmental dynamics that govern these empirical observations and suggest novel prevention strategies that may be tested experimentally.
Subject(s)
Neoplasms/etiology , Adaptation, Physiological , Animals , Cell Proliferation , Disease Progression , Humans , Models, BiologicalABSTRACT
OBJECTIVE: We have shown previously that cutting or loading articular cartilage resulted in a fibroblast growth factor-2 (FGF-2) dependent activation of the extracellularly regulated kinase (ERK), and induction of a number of chondrocyte regulatory proteins including tissue inhibitor of metalloproteinase-1 and matrix metalloproteinases 1 and 3. An extracellular matrix-bound pool of FGF-2 was apparent, which could be liberated from the tissue by heparitinase (Vincent et al., Proc Natl Acad Sci U S A 2002;99(12):8259-64, Vincent et al., Arthritis Rheum 2004 Feb;50(2):526-33). Our objectives were to determine where FGF-2 was stored in articular cartilage, to which proteoglycan it was bound, and to elucidate its role in chondrocyte mechanotransduction. METHODS: Immunohistochemistry and confocal microscopy were used to localise FGF-2 in the tissue. In vitro binding studies were performed using IASYS surface plasmon resonance. To study the role of pericellular FGF-2 in mechanotransduction cartilage explants or articular chondrocytes encapsulated in alginate were loaded using an in house loading rig. The loading response was assessed by the activation of ERK, in the presence or absence of a specific FGFR inhibitor. RESULTS: Here we have identified perlecan as the heparan sulphate proteoglycan that sequesters FGF-2 in articular cartilage. Perlecan and FGF-2 co-localised within the type VI collagen-rich pericellular matrix of porcine and human articular cartilage. Chondrocytes encapsulated in alginate were able to accumulate pericellular perlecan and FGF-2 in culture, and deliver an FGF-dependent activation of ERK when loaded. CONCLUSION: Loading-induced ERK activation was dependent upon the presence and concentration of pericellular FGF-2, suggesting a functional role for this matrix-bound growth factor in chondrocyte mechanotransduction.
Subject(s)
Cartilage, Articular/physiology , Chondrocytes/physiology , Extracellular Matrix/physiology , Fibroblast Growth Factor 2/physiology , Heparan Sulfate Proteoglycans/physiology , Mechanotransduction, Cellular/physiology , Animals , Cartilage, Articular/metabolism , Chondrocytes/metabolism , Extracellular Matrix/metabolism , Fibroblast Growth Factor 2/metabolism , Heparan Sulfate Proteoglycans/metabolism , SwineABSTRACT
Previous work has demonstrated that for games defined by differential or difference equations with a continuum of strategies, there exists a G-function, related to individual fitness, that must take on a maximum with respect to a virtual variable v whenever v is one of the vectors in the coalition of vectors which make up the evolutionarily stable strategy (ESS). This result, called the ESS maximum principle, is quite useful in determining candidates for an ESS. This principle is reformulated here, so that it may be conveniently applied to matrix games. In particular, we define a matrix game to be one in which fitness is expressed in terms of strategy frequencies and a matrix of expected payoffs. It is shown that the G-function in the matrix game setting must again take on a maximum value at all the strategies which make up the ESS coalition vector. The reformulated maximum principle is applicable to both bilinear and nonlinear matrix games. One advantage in employing this principle to solve the traditional bilinear matrix game is that the same G-function is used to find both pure and mixed strategy solutions by simply specifying an appropriate strategy space. Furthermore we show how the theory may be used to solve matrix games which are not in the usual bilinear form. We examine in detail two nonlinear matrix games: the game between relatives and the sex ratio game. In both of these games an ESS solution is determined. These examples not only illustrate the usefulness of this approach to finding solutions to an expanded class of matrix games, but aids in understanding the nature of the ESS as well.