Johannes Thaler

Cihan Ay

Harald Weinstabl

Daniela Dunkler

Ralph Simanek

Rainer Vormittag

Jean-Marie Freyssinet

Christoph Zielinski

Dr Ingrid Pabinger
email: ingrid.pabinger@meduniwien.ac.at

Circulating procoagulant microparticles in cancer patients

Keywords: microparticles, cancer, venous thromboembolism, prothrombinase assay CATS, Cancer and Thrombosis Study, CI, confidence interval, MPs, microparticles, HR, hazard ratio, PFP, platelet-free plasma, nM PS, nanomolar phosphatidylserine equivalent, PS, phosphatidylserine, PSGL-1, P-selectin glycoprotein ligand, TF, tissue factor, VTE, venous thromboembolism

Accumulating evidence indicates that microparticles (MPs) are important mediators of the interaction between cancer and the hemostatic system. We conducted a large prospective cohort study to determine whether the number of circulating procoagulant MPs is elevated in cancer patients and whether the elevated MP levels are predictive of occurrence of venous thrombembolism (VTE). We analyzed plasma samples of 728 cancer patients from the ongoing prospective observational Vienna Cancer and Thrombosis Study (CATS). Study endpoint was the occurrence of symptomatic VTE. Sixty-five age-and sex-matched healthy controls were recruited for defining the cut-off point for elevated MPs (4.62 nanomolar phosphatidylserine nM PS ), which was set at the 95th percentile of MP levels in healthy controls. The measurement of MPs was performed after capture onto immobilized annexin-V, and determination of their procoagulant activity was quantified with a prothrombinase assay. During a median observation period of 710 days 53 patients developed VTE. MP levels (nM PS) were significantly higher in cancer patients than in healthy controls (median 25th-75th percentile : 3.95 1. 74-7.96 vs. 1.19 0.81-1.67 , p<0.001). Multivariate analysis including age, sex, surgery, chemo-and radiotherapy showed no statistically significant association of the hazard ratio of elevated MPs with VTE (0.95 [95% CI: 0.55-1.64], p=0.856). In conclusion, MP levels were elevated in cancer patients compared to healthy individuals in this study. However, elevated MP levels were not predictive of VTE.

Introduction

Circulating procoagulant microparticles (MPs) are phosphatidylserine (PS)-rich, small vesicles with a diameter of 0.1 m to 1 m, which are emitted from cell membranes into the blood when cells are activated or undergo apoptosis. The majority of MPs in plasma are derived from platelets [START_REF] Siljander | Platelet-derived microparticles associate with fibrin during thrombosis[END_REF], erythrocytes [START_REF] Willekens | Erythrocyte vesiculation: A selfprotective mechanism?[END_REF], endothelial cells [START_REF] Sabatier | Interaction of endothelial microparticles with monocytic cells in vitro induces tissue factor-dependent procoagulant activity[END_REF], lymphocytes [START_REF] Diamant | Cellular microparticles: New players in the field of vascular disease?[END_REF], monocytes [START_REF] Satta | The significance of human monocyte thrombomodulin during membrane vesiculation and after stimulation by lipopolysaccharide[END_REF] and smooth muscle cells [START_REF] Schecter | Release of active tissue factor by human arterial smooth muscle cells[END_REF]. An important role of MPs in the hemostatic system has been proposed, because MPs provide a negatively charged surface for the aggregation and subsequent activation of coagulation factors, and promote coagulation via the expression of coagulation proteins and adhesion molecules on their surface [START_REF] Hugel | Membrane microparticles: Two sides of the coin[END_REF][START_REF] Falati | Accumulation of tissue factor into developing thrombi in vivo is dependent upon microparticle p-selectin glycoprotein ligand 1 and platelet p-selectin[END_REF][START_REF] Conde | Microvesicle-associated tissue factor and trousseau's syndrome[END_REF][START_REF] Zwicker | Tumor-derived tissue factor-bearing microparticles are associated with venous thromboembolic events in malignancy[END_REF]. Experimental studies have demonstrated that MPs are involved in the pathophysiology of thrombosis [START_REF] Morel | Procoagulant microparticles: Disrupting the vascular homeostasis equation?[END_REF]. It has been shown that a subgroup of MPs are guided to a developing thrombus through the interaction of P-selectin glycoprotein ligand-1 (PSGL-1), which is present on MPs, and P-selectin, which is expressed on activated platelets and endothelial cells.

The accumulation of MP-bound tissue factor (TF) further supports thrombus growth [START_REF] Polgar | The p-selectin, tissue factor, coagulation triad[END_REF].

Clinical studies have reported elevated MP levels in different thrombotic disorders [START_REF] Morel | Procoagulant microparticles: Disrupting the vascular homeostasis equation?[END_REF]. Yet, results are conflicting regarding the clinical relevance of MPs in venous thromboembolism (VTE).

While highly elevated MP levels were measured shortly after an acute VTE event in cancer and non-cancer patients [START_REF] Tesselaar | Microparticle-associated tissue factor activity: A link between cancer and thrombosis?[END_REF][START_REF] Chirinos | Elevation of endothelial microparticles, platelets, and leukocyte activation in patients with venous thromboembolism[END_REF] MP levels were not elevated 3 months after a VTE event in a high-risk population of non-cancer patients with a history of recurrent VTE [START_REF] Ay | Circulating procoagulant microparticles in patients with venous thromboembolism[END_REF]. Although cancer patients have a 4-to 7-fold increased risk of VTE and VTE is the second leading cause of death in this population, after cancer itself [START_REF] Heit | Risk factors for deep vein thrombosis and pulmonary embolism: A population-based case-control study[END_REF][START_REF] Blom | Malignancies, prothrombotic mutations, and the risk of venous thrombosis[END_REF][START_REF] Sousou | Identifying cancer patients at risk for venous thromboembolism[END_REF] large prospective clinical studies investigating the predictive role of MPs in cancer-associated VTE are still lacking.

As the pathophysiology of cancer-associated VTE is not fully understood and laboratory parameters to predict cancer-associated VTE are scarce, attention in recent research activities has been focused on MPs as mediators of the interaction between cancer and the hemostatic system.

Accumulating evidence indicates that MPs contribute to the development of VTE in cancer patients [START_REF] Hugel | Membrane microparticles: Two sides of the coin[END_REF]. High levels of circulating MPs were found in breast cancer [START_REF] Toth | Platelet-derived microparticles and coagulation activation in breast cancer patients[END_REF], gastric cancer [START_REF] Kim | Elevated levels of circulating platelet microparticles, vegf, il-6 and rantes in patients with gastric cancer: Possible role of a metastasis predictor[END_REF], and pancreatic cancer patients [START_REF] Tesselaar | Microparticle-associated tissue factor activity: A link between cancer and thrombosis?[END_REF] compared to healthy individuals, and TF-positive MPs were postulated as candidate biomarkers for the development of cancer-associated VTE [START_REF] Sud | Cancer-associated thrombosis: Risk factors, candidate biomarkers and a risk model[END_REF][START_REF] Langer | Tissue factor procoagulant activity of plasma microparticles in patients with cancer-associated disseminated intravascular coagulation[END_REF].Therefore, the aim of this prospective observational cohort study was to determine, whether baseline levels of MPs are elevated in cancer patients with different tumour types compared to sex-and age-matched healthy individuals and whether elevated MP levels have a predictive role in the occurrence of VTE in prospectively followed cancer patients.

Design and Methods

Study population

The study protocol was approved by our local Ethics Committee. Informed consent was signed before patients and healthy individuals were included in this study. This study was designed as a case-control study as part of the ongoing observational Vienna Cancer and Thrombosis Study (CATS), which was initiated in 2003. First data of this study were published in 2008 and 2009 [START_REF] Ay | High plasma levels of soluble p-selectin are predictive of venous thromboembolism in cancer patients: Results from the vienna cancer and thrombosis study (cats)[END_REF][START_REF] Vormittag | High factor viii levels independently predict venous thromboembolism in cancer patients: The cancer and thrombosis study[END_REF][START_REF] Ay | D-dimer and prothrombin fragment 1 + 2 predict venous thromboembolism in patients with cancer: Results from the vienna cancer and thrombosis study[END_REF]. Plasma samples of 728 cancer patients with newly diagnosed cancer or progression of disease after complete or partial remission, including malignancies of the brain, breast, lung, stomach, colon, pancreas, kidney, prostate, hematological malignancies (multiple myeloma, high and low grade lymphoma) or other sites (mainly of the gynecologic system and sarcoma) were analyzed. Main patients' characteristics are listed in table 1. At study inclusion, patients underwent a structured interview on their medical history and data on the site of tumor, histology and tumorstage were documented. Patients were followed prospectively and information on the course of disease and on the occurrence of VTE was collected for every patient at three-month intervals.

Patients were instructed to contact the Division of Haematology and Haemostaseology (Vienna General Hospital), if they perceived symptoms indicative of VTE (deep vein thrombosis or pulmonary embolism), stroke, myocardial infarction, or peripheral arterial thrombosis. In case of death, data on the time and the cause of death including autopsy protocols were collected. Each patient was followed over a two-year period or until the occurrence of VTE, death or loss of follow-up.

We recruited 65 age-and sex-matched healthy control subjects without a history of arterial or venous thrombosis and cancer from the same geographic region and ethnic backgrounds as our patients. Based on their MP levels we defined the cut-off point for elevated MPs, which was set at the 95 th percentile of the MP levels in our healthy control group.

Blood sampling and laboratory methods

Venous blood samples were drawn into citrate vacuum tubes (Vacuette, Greiner-Bio One, Kremsmuenster, Austria) by atraumatic and sterile antecubital venepuncture on the day of study entry. The citrated blood was centrifuged (Rotanta/TRC, Hettich, Tuttlingen, Germany) at 1500 g for 15 minutes to get platelet-poor plasma. Platelet-free plasma (PFP) was obtained by a second centrifugation step (Eppendorf, Hamburg, Germany) at 13400 g for two minutes. The centrifugation of each sample was performed within 1 hour after blood sampling and the freezing of each sample within 1 hour after centrifugation. Plasma aliquots were stored at minus 80°C until assays were performed in series.

MPs were captured by immobilized annexin V as previously described [START_REF] Aupeix | The significance of shed membrane particles during programmed cell death in vitro, and in vivo, in hiv-1 infection[END_REF]. In brief, biotinylated annexin V (annexin V Bi ; Roche Diagnostics, Mannheim, Germany) was insolubilized onto streptavidin-coated microtitration plates (Roche Diagnostics, Mannheim, Germany). The plates were washed 3 times with TBS-Ca 2+ (50 mM Tris buffer, pH 7.5, containing 120 mM NaCl, 2.7 mM KCl, 1 mM CaCl 2 ). 300 l aliquots of PFP were thawed and thrombin inhibitor (30 M PPACK; Merck, Darmstadt, Germany) and factor Xa inhibitor (30 M 1.5-Dns-GGACK; Merck, Darmstadt, Germany) were added. We then incubated 100 l PFP/well for 30 minutes at 37 C.

Each sample was measured in duplicate. After 4 washing steps, the anionic phospholipid content was determined by a prothrombinase assay. In this assay immobilized MPs were incubated in a final volume of 150 l with factor Va (250 pM; American Diagnostica, Stamford, USA), Xa (9.3 pM; Hyphen BioMed, Andresy, France), prothrombin (0.7 M; Hyphen BioMed, Andresy, France) and CaCl 2 (2.3 mM) for 15 minutes at 37 C. Then Chromozym TH (1.5 mM; Roche Diagnostics, Mannheim, Germany), a chromogenic substrate for thrombin, was added and the solution was incubated for 4 minutes. The chromogenic substrate was cleaved by thrombin.

Change of colour was measured photometrically. Linear absorbance changes were recorded at 405 nm using a microtitration plate reader (MR 7000, Dynatech, Guernsey, Great Britain).

In the method first described by Aupeix et al [START_REF] Aupeix | The significance of shed membrane particles during programmed cell death in vitro, and in vivo, in hiv-1 infection[END_REF] a kinetic software was used for data analysis. To make this method applicable on a standard laboratory photometer, the colour reaction was stopped after 4 minutes by the addition of EDTA (5 mM). As the colour development reaction followed a Michaelis Menten kinetic, we compared our measurements to standard curves by means of nonlinear regression analysis. Results were expressed as nanomolar phosphatidylserine equivalent (nM PS) by reference to a standard curve that we had constructed by using phospholipids of defined composition (33% PS and 67% phosphatidylcholine, PC) as described by Pigault et al [START_REF] Pigault | Formation of two-dimensional arrays of annexin v on phosphatidylserine-containing liposomes[END_REF]. Phospholipids were quantified by the Steward-assay [START_REF] Stewart | Colorimetric determination of phospholipids with ammonium ferrothiocyanate[END_REF]. When MPs were present, the reaction was significantly accelerated because of the high amount of PS present on MPs.

Statistical analysis

Continuous variables were described with the median and the 25th-75th percentile. The median follow-up time was calculated with the reverse Kaplan Meier method. A Wilcoxon test was applied to test differences in the distribution of MPs between cancer patients and controls. Elevated levels of MPs were defined as those exceeding the 95th percentile of healthy controls.

Kaplan Meier analysis and a log-rank test were used to compare the cumulative probability of VTE between cancer patients with elevated or non-elevated MPs.

Univariate and multivariable Cox regression analysis were applied to calculate the risk of VTE from study inclusion until first thrombosis. Data were censored at last follow-up, patient's death or at end of follow up after 2 years. The multivariable Cox regression analysis comprised the following parameters: MPs, age, sex, surgery, chemo-and radiotherapy. The covariate of main interest was elevated MPs. We assumed that surgery, chemotherapy and radiotherapy would not only modify the VTE risk at the exact time-point of the procedure, but also for a certain timeperiod beyond: 6 consecutive weeks for surgery and 4 for chemo-or radiotherapy (from the first day of a treatment cycle until the last day plus 4 weeks). Time-dependent effects on the VTE risk were modelled by inclusion of these three time-dependent dichotomous variables into the multivariable Cox regression model. Furthermore, the analysis was adjusted for age at study inclusion and sex. The multivariable Cox regression model was tested for all pair-wise interactions and interactions with log(time) by means of candidate variables. As no significant interaction was found (p-value smaller than 0.01), no interaction was added.

A p-value smaller than 0.05 was regarded as statistically significant. All calculations were conducted with SAS 9.2.

Results

Circulating procoagulant MP in patients and controls

In this study, plasma levels of circulating procoagulant MPs were measured in 728 patients with solid and hematological malignancies and in 65 healthy individuals. The baseline characteristics of patients and controls are given in table 1. MP levels (nanomolar phosphatidylserine nM PS ) were significantly higher in cancer patients than in healthy controls (3.95 nM PS 1.74-7.96 vs. 1.19 nM PS 0.81-1.67 , p<0.001, figure 1). The cut-off point for elevated MPs was set at 4.62 nM PS, which represented the 95th percentile of MP levels in the control individuals of our study. Elevated MPs were found in 327 (44.9%) cancer patients.

Association of circulating MPs with the occurrence of VTE in cancer patients

During a median observation period of 710 days (25th-75th precentile: 262-731) 53 (7.3%) patients developed VTE. The overall cumulative probability of VTE in cancer patients was 5.9% after 6 months and 7.3% after one year. The characteristics of patients who developed VTE are given in table 2.

There was no statistically significant difference of MP levels between cancer patients with localized disease and cancer patients with distant metastasis at study inclusion (4.21 nM PS 1.93-7.71 vs. 3.67 1.61-4.30 , p=0.255). MP levels did also not differ significantly between cancer patients who had developed VTE compared to those who had not developed VTE by the time of follow-up (3.56 nM PS 1.86-7.42 vs. 3.97 nM PS 1.73-7.97 , p=0.846). For further statistical analysis cancer patients were dichotomized into those with elevated and those with non-elevated MP levels (cut-off point 4.62 nM PS, representing the 95th percentile of MP levels in healthy controls). In univariate Cox regression analysis the hazard ratio (HR) of elevated circulating MPs ( 4.62 nM PS) was not associated with VTE (HR 0.99 [95% CI: 0.58-1.70], p=0.968). In multivariable analysis, after additionally adjusting for age, sex, surgery, chemo-and radiotherapy the HR of elevated MPs was 0.95 ([95% CI: 0.55-1.64], p=0.856). Additional multivariable analyses including different tumour subgroups, soluble P-selectin, platelet and leukocyte counts did not reveal a significant association of elevated circulating MPs with the occurrence of VTE (data not shown).

In patients with elevated MP levels cumulative probabilities of VTE were 5.2% after 6 months and 6.8% after 1 year and in those with non-elevated MP levels they were 6.4% after 6 months and 7.7% after 1 year, respectively (Log-rank test: p=0.968).

Discussion

In this study we measured circulating procoagulant MPs in 728 cancer patients, who were followed prospectively for a median of 710 days, and in a control group of age-and sex-matched healthy individuals. In line with previous reports MP levels at study inclusion were significantly higher in the group of cancer patients than in healthy controls without cancer [START_REF] Kim | Elevated levels of circulating platelet microparticles, vegf, il-6 and rantes in patients with gastric cancer: Possible role of a metastasis predictor[END_REF][START_REF] Toth | Platelet-derived microparticles and coagulation activation in breast cancer patients[END_REF][START_REF] Tesselaar | Microparticle-associated tissue factor activity: A link between cancer and thrombosis?[END_REF][START_REF] Hron | Tissue factor-positive microparticles: Cellular origin and association with coagulation activation in patients with colorectal cancer[END_REF]. MPs were also analyzed for their potential of being a predictive parameter for the occurrence of VTE in cancer patients. However, MP levels were not found to be predictive of VTE in cancer patients.

Consistent with previous studies, in our study we observed the highest VTE rates in patients with brain tumors (high grade glioma) [START_REF] Kakkar | Venous thrombosis in cancer patients: Insights from the frontline survey[END_REF]. However, we did not find higher MP levels in the subgroup of patients with brain tumors compared to other tumor entities. The high risk of VTE in brain tumors is not fully understood. VTE is a multifactorial disease and in cancer patients several risk factors such as immobilization, tumor subtype, tumor size, chemotherapy or radiotherapy and many other acquired or inherited VTE risk factors contribute to the very high VTE risk. This may be especially true for brain tumor patients in whom the high VTE risk exceeds the postoperative period and stays high throughout the course of disease. [START_REF] Jenkins | Venous thromboembolism in malignant gliomas[END_REF][START_REF] Simanek | Venous thromboembolism and survival in patients with high-grade glioma[END_REF][START_REF] Semrad | Epidemiology of venous thromboembolism in 9489 patients with malignant glioma[END_REF][START_REF] Rong | Pten and hypoxia regulate tissue factor expression and plasma coagulation by glioblastoma[END_REF].

Up to date, several methods for the measurement of MPs have been developed in order to quantify circulating MPs. Most studies used either flow cytometry to measure the number of MPs, or functional chromogenic assays were deployed to determine the procoagulant potential of MPs [START_REF] Jy | Measuring circulating cell-derived microparticles[END_REF]. In our large prospective study we used a functional chromogenic assay in a solid phase for MP detection. For the construction of standard curves phospholipids of defined composition were used (33% PS and 67% PC). Interestingly, recent studies showed that the phospholipid composition of circulating procoagulant MPs varies depending on type of originating cell and type of stimulation [START_REF] Hussein | Phospholipid composition of in vitro endothelial microparticles and their in vivo thrombogenic properties[END_REF][START_REF] Biro | The phospholipid composition and cholesterol content of platelet-derived microparticles: A comparison with platelet membrane fractions[END_REF][START_REF] Weerheim | Phospholipid composition of cell-derived microparticles determined by one-dimensional high-performance thin-layer chromatography[END_REF]. Therefore, great variations of the PS and PC distribution on the surface of circulating MPs of different origin may be possible. Especially tumor derived MPs may have a completely different phospholipid composition as the phospholipids used in our standard curves.

Nevertheless, this prothrombinase assay is a well established method that had previously been used in several different clinical studies for the assessment of circulating procoagulant MPs in patients with a variety of diseases, such as sepsis [START_REF] Nieuwland | Cellular origin and procoagulant properties of microparticles in meningococcal sepsis[END_REF], HIV [START_REF] Aupeix | The significance of shed membrane particles during programmed cell death in vitro, and in vivo, in hiv-1 infection[END_REF], type 1 and type 2 diabetes [START_REF] Sabatier | Type 1 and type 2 diabetic patients display different patterns of cellular microparticles[END_REF], acute coronary syndrome [START_REF] Mallat | Elevated levels of shed membrane microparticles with procoagulant potential in the peripheral circulating blood of patients with acute coronary syndromes[END_REF], VTE [START_REF] Chirinos | Elevation of endothelial microparticles, platelets, and leukocyte activation in patients with venous thromboembolism[END_REF][START_REF] Manly | Increased microparticle tissue factor activity in cancer patients with venous thromboembolism[END_REF], and preeclampsia [START_REF] Gonzalez-Quintero | Elevated plasma endothelial microparticles in preeclampsia[END_REF]. Yet, up to date there are no studies applying this functional prothrombinase assay in a solid phase in cancer. Using this method we detected higher MP levels in cancer patients than in healthy individuals, but the difference between MP levels of cancer patients who developed future VTE and those who did not was not statistically significant.

As freeze-thawing cycles might increase the number of circulating MPs [START_REF] Valeri | Freezing human platelets with 6 percent dimethyl sulfoxide with removal of the supernatant solution before freezing and storage at -80 degrees c without postthaw processing[END_REF], a limitation of the study might be that we used frozen plasma samples for determination of MPs. However, samples of cancer patients and controls were freshly thawed and treated equally and different preanalytical conditions can be excluded.

Recently, new assays for the measurement of TF-positive MPs or MP-associated TF activity have been established that so far have been predominantly performed in cancer patients. Hron et al measured two-fold higher levels of TF positive MPs in colorectal cancer patients compared to healthy controls and reported that TF positive MPs correlated with elevated D-dimer levels, which is a marker for activation of coagulation [START_REF] Hron | Tissue factor-positive microparticles: Cellular origin and association with coagulation activation in patients with colorectal cancer[END_REF]. In two small studies functional chromogenic assays were used to compare MP-associated TF activity in cancer with and without acute VTE.

Tesselaar et al measured an elevated MP-associated TF activity in metastatic breast cancer patients and metastatic pancreatic cancer patients who presented with acute VTE compared to cancer patients without VTE [START_REF] Tesselaar | Microparticle-associated tissue factor activity: A link between cancer and thrombosis?[END_REF]. Strikingly, no MP-associated TF activity was found in non-cancer patients who had suffered from acute VTE. Also Manly et al detected higher MP-associated TF activity levels in 53 cancer patients with different tumour types and acute VTE than in 13 cancer patients without VTE [START_REF] Manly | Increased microparticle tissue factor activity in cancer patients with venous thromboembolism[END_REF]. Furthermore, Zwicker et al applied a newly developed method, the impedance flow cytometry, and showed elevated numbers of TF-positive MPs in 60% of cancer patients with acute VTE compared to 27% of cancer patients without VTE [START_REF] Zwicker | Tumor-derived tissue factor-bearing microparticles are associated with venous thromboembolic events in malignancy[END_REF].In contrast to the above-mentioned studies, in our large prospective study we measured baseline MP levels of cancer patients and followed them over a 2-year period or until occurrence of VTE or death or loss of follow-up, in order to find parameters that might be predictive of cancer-associated VTE.

According to our study protocol, cancer patients with acute VTE at least 3 months prior to study inclusion were not enrolled in order to exclude the effect of an acute VTE event on MP levels and other laboratory parameters. In our study MPs were quantified by measuring the PS content on the MP surface after capture onto immobilized annexin V and by determining their procoagulant activity with a prothrombinase assay. PS was reported to be a highly procoagulant phospholipid [START_REF] Morel | Procoagulant microparticles: Disrupting the vascular homeostasis equation?[END_REF], but our results do not support a predictive role of PS-exposing MPs for the occurrence of VTE in cancer patients. Whether MPs that additionally express high levels of TF on their surface might be predictive of cancer-associated VTE, remains to be further elucidated in large prospective studies.

In conclusion, we found that MP levels in cancer patients were elevated compared to healthy individuals. However, with the prothrombinase assay we did not find a significant association between circulating procoagulant MPs and the risk of developing VTE in cancer patients. It has to be considered, that VTE is a complex multifactorial process and that a plethora of pathologic changes occurs in the hemostatic system of cancer patients. The role of procoagulant MPs in cancer-associated VTE still needs to be addressed in other large prospective studies. These studies should also focus on the measurement of TF-positive MPs as a predictive parameter, because recent experimental and clinical research indicates a potential role of this specific MP subpopulation in the pathogenesis of cancer-associated VTE. Open circles represent outliers.

Figure 1 .

 1 Figure 1. Box plots: Microparticles in healthy controls and cancer patients

Table 1 .

 1 Baseline characteristics of total study population

	Characteristics of all patients (n=728)
	Age at study entry, median; (IQR), yrs	62 (53-68)
	Gender
	Female, n (%)	341 (46.84)
	Male, n (%)	387 (53.16)
	Site of cancer	n (%)
	Brain	86 (11.8)
	Breast	125 (17.2)
	Lung	102 (14.0)
	Stomach	32 (4.4)
	Colon	102 (14.0)
	Pancreas	43 (5.9)
	Kidney	21 (2.9)
	Prostate	80 (11.0)
	Multiple Myeloma	17 (2.3)
	Lymphoma	74 (10.2)
	Others	46 (6.3)
	Characteristics of all controls (n=65)
	Age at study entry, median; (IQR), yrs	60 (58-66)
	Gender
	Female, n (%)	30 (46.2)
	Male, n (%)	35 (53.8)

Table 2 .

 2 Event characteristics of cancer patients with VTE n=53

	Site of thrombotic event

Acknowledgements: We thank all persons that supported us in patient recruitment for the Vienna Cancer and Thrombosis Study (CATS), Silvia Koder for skillful technical assistance and Tanja Altreiter for proof-reading of the manuscript.

Financial support: This study was supported by grants from the "Jubiläumsfonds" of the Austrian National Bank, by an unrestricted grant from Pfizer Austria, and the "Fellinger Krebsforschung".


Philip Savarimuthu

Sunil Anthony

Sylvia M Varughese

Draper
email: smdraper@tcd.ie

Impact of molecular structure on intermolecular interactions and organic solid state luminescence in supramolecular systems

Keywords: Organic solid state luminescence, Co-crystals, Supramolecular luminescent systems, Supramolecular interactions

come

Impact of molecular structure on intermolecular interactions and organic luminescence in supramolecular systems

Introduction

Solid state organic luminescent materials particularly with tunable luminescence are of great interest in recent years, because of their applications in the field of optoelectronic devices, including organic electroluminescence (EL), light-emitting diodes (LEDs), and optical sensors. [1] Organic molecules for example π-conjugated molecules show high luminescence in dilute solution [2] but become weakly luminescent in the solid state due to the formation of less emissive species such as excimers and restrict the applications of these molecules in high density optical systems. [3] Strategies, such as the introduction of bulky substituents or the synthesis of propeller-shaped luminescent molecules which prevent close-packing in the solid state, have successfully lead to organic molecules that luminesce in the solid state. [4] The propeller-shaped triphenylamine and its derivatives often show intense solid state luminescence and have been used in organic LEDs and electroluminescent devices. [5] The tuning of organic, solid state luminescence is mostly achieved by the modification of substitutions on single molecules [6] or the exploitation of polymorphism. [7] The latter approach is effective as the optical properties in the solid are The molecular arrangement of luminophores can be controlled through supramolecular interactions and this in turn can be used to tune or enhance the solid state luminescence. [START_REF] Mizobe | [END_REF] Although weak in organic compounds, supramolecular interactions such as H-bonding influence the energy levels of organic compounds and the stability of the excited state depending on the orientation of neighboring molecules. [8] As an example, the supramolecular interactions of an organic luminescent acid has been effectively controlled by mixing it with amines and tuned the solid state luminescence. [10] In general, deprotonation and protonation provides a simple strategy to switch or tune across a wide wavelength range, the emission of organic luminescent acids. We have recently reported the gradual shift of solid state luminescence from red to blue by utilizing unusual and subtle variations in the intermolecular interactions of 2-cyano-3(4-(diphenylamino)phenyl) acrylic acid (CDPA) with amines. [11] The supramolecular

Results and Discussion

CDPA was synthesized by using the reported procedure 12 Replacing the H atom of the para carbon with a methyl group (4-picoline (pKa = 5.99), 1) leads to the formation of supramolecular structure with usual pyridine-carboxylic acid intermolecular interactions (Fig. 2). [13] 1 forms with 1:1 molecule ratio of CDPA and 4-picoline. similarly unexpected deprotonation of CDPA on using morpholine, the weakest base of the alicyclic amines used (piperidine, pyrrolidine and morpholine) in the preparation of supramolecular luminescent systems. [11] As noted previously, there is a complex balance of forces at play in the solid state structures which go beyond or over-ride mere solutionbased pKa considerations.

Powder X-ray diffraction (PXRD) patterns of 1-3 were measured and compared with the simulated powder pattern from single crystal data to confirm the purity of the sample (Fig. 5). The PXRD patterns of 1 and 2 are essentially the same as their respective simulated patterns (Fig. S1 and S2 in the supporting information) which indicates that the samples are not having polymorphism. A different PXRD pattern of 3 compared to CDPA, 1 and 2 suggests a different molecular arrangement in 3.

The solid state luminescence spectra of supramolecular luminescent systems of 1-3 are shown in figure 6. CDPA exhibits strong luminescence in CH 2 Cl 2 solutions with λ max at 603 nm and estimated quantum yield (Ф f ) 0.165 (on comparison with coumarin 6).

Solid CDPA has similar luminescence intensity. [11] Complexation with pyridine and picoline also show similar luminescence intensity compared to CDPA solid (Fig. S3 in the supporting information). However, the different supramolecular structures observed in the structural analysis were expected to lead to tuning of solid state luminescence.

Powdered and deprotonated CDPA (Fig. 3) are more similar to the CDPA-pyridine supramolecular systems and led to solid state luminescence at 559 nm. 3 shows solid state luminescence at λ max 585 nm, a similar to that of CDPA powder, but the spectrum of 3 is sharper and narrower than that observed in the broad spectrum of CDPA.

Further information on the nature of the excited states was obtained by measuring the fluorescence lifetimes of CDPA in CH 3 CN, CDPA and 1-3 powdered materials.

Decays were monitored at the corresponding emission maxima.

Experimental Section

Synthesis and characterization:

Triphenylamine, cyanoacetic acid, POCl 3 , anhydrous dimethyl formamide (99.8%), pyridine, 2-picoline, 3-picoline and 4-picoline were obtained from Aldrich and used as received. CDPA was prepared following the reported procedure. [12] Elemental analyses were measured with a Perkin-Elmer 2400 II CHN analyzer.

General procedure for 1 -3 preparation:

A mixture of CDPA (100 mg, 0.29 mmol) and amine (0.3 mmol) was dissolved in acetonitrile (15 ml) and left to stand at room temperature. Crystals of 1-3 form after near complete evaporation of solvent. Above 90 % yield was obtained for 1-3. X-ray Crystallography: PXRD measurements were recorded using Siemens diffraktometer-D500 at room temperature.

Elemental analysis

Single crystals were carefully chosen after they were viewed through a polarizing microscope. The crystals were glued to a thin glass fiber using an adhesive (cyano acrylate) and mounted on a diffractometer equipped with an APEX CCD area detector.

The data collection was carried out at 150K and no extraordinary methods were employed, except that the crystals were smeared in NIH immersion oil to protect them from ambient laboratory conditions. The intensity data were processed using Bruker's suite of data processing programs (SAINT), and absorption corrections were applied using SADABS. [START_REF] Sheldrick | Area Detector Correction[END_REF] The structure solution of all the complexes was carried out by direct methods, and refinements were performed by full-matrix least-squares on F 2 using the SHELXTL-PLUS [15] suite of programs. All the structures converged to good R factors.

All the non-hydrogen atoms were refined anisotropically, and the hydrogen atoms were fixed on calculated position using appropriate HFIX options in Shelxtl and were refined isotropically.

Crystallographic data for

  organization but it offers little predictability and the former requires interactive synthetic improvement.

  luminescent host system of CDPA-piperidine exhibited CH 3 CN selective luminescence switching. In this manuscript, we report the effect of molecular structure on the supramolecular interactions and optical properties of CDPA supramolecular luminescent systems with substituted pyridines. Supramolecular luminescence systems based on CDPA and 4-picoline (1), 3-picoline (2), and 2-picoline (3) are presented. The change of methyl position in the pyridine leads to different molecular arrangements of the CDPA in the solid state and a variety of supramolecular interactions. Most importantly, the change of CDPA molecular arrangement in the solid state is accompanied by changes in the solid state luminescence.

  The 4-picoline nitrogen atoms and ortho carbon form N…H-O and O…H-C intermolecular interactions with the carboxylic acid of CDPA in the crystal lattice of 1. The same carbonyl oxygen atom further interacts with the other ortho carbon of another 4-picoline molecule and forms O…H-C interactions. The oxygen atom of the carboxylic hydroxyl group is involved in the O…H-C interactions with the pyridine meta carbon along the z-axis. The N…H-O and O…H-C intermolecular interactions, lead to the formation of a sheet-like supramolecular structure in the xz plane. The single crystal X-ray analysis of 2 (3-picoline, pKa= 5.70) reveals another interesting supramolecular structure. 2 forms with a 2:1 molecule ratio of CDPA and 3picoline molecule in the crystal lattice. The unit cell contains one molecule of 3-picoline for every two molecules of CDPA. Of the two CDPA molecules only one is deprotonated. In 2, one of the deprotonated CDPA oxygen atoms forms O…H-N intermolecular H-bond interactions with 3-picoline and the other oxygen rather than forming any usual intermolecular H-bond interactions with the H atom of the ortho carbons of pyridine, forms O…H-O interactions with a protonated CDPA carboxylic acid (Fig.3). The carbonyl oxygen of protonated CDPA forms O…H-C interactions with the H atom of the para carbon of 3-picoline. The ortho carbon and the methyl carbon of 3-in the C-H…N interactions with a cyano group of a different deprotonated CDPA molecule. The cyano group of protonated CDPA did not form any supramolecular interactions. These multiple intermolecular H-bond interactions in 2 lead to the formation of a one-dimensional network structure along the y-axis. Despite repeated attempts, we could not obtain good quality single crystals of CDPA-2-picoline (Fig.4, pKa = 6.00, 3) from any solvent. It forms flake-like crystalline products from CH 3 CN. The deprotonation of CDPA in 2 is rather surprising on consideration of the low basicity of 3-picoline, the weakest base among the picolines used. We have also observed

  CDPA shows solid state luminescence at 587 nm. The subtle change of carboxylic acid H-bond interactions from O-H … NC in CDPA to O-H … O-H (Fig. 1) in pyridine blue shifts the luminescence from 587 nm to 565 nm. The usual pyridinecarboxylic acid H-bond formation in 1 (Fig. 2) further blue shifts the luminescence to 548 nm. 2 which has a deprotonated CDPA molecule in the crystal lattice should have further solid state luminescence but the presence of a protonated CDPA molecule and the formation of O…H-O intermolecular interactions between protonated

  Figure Captions

Figure 2 :

 2 Figure 2: Crystal packing of 1 (a) and selected H-bond interactions in the crystal lattice (b). Only H atoms involved in H-bond interactions are shown; C (grey), N (blue), O (red), H (white); H-bonds (broken line). d H … A distances (Å) are marked. Molecular structure of 1 with H-bond interactions formed in the crystal lattice (c).

Figure 3 :

 3 Figure 3: Crystal packing of 2 (a) and selected H-bond interactions in the crystal lattice (b). Only H atoms involved in H-bond interactions are shown; C (grey), N (blue), O (red), H (white); H-bonds (broken line). d H … A distances (Å) are marked. Molecular structure of 2 with H-bond interactions formed in the crystal lattice(c).

Figure 4 :

 4 Figure 4: Molecular structure of 3.

  Figure 5: PXRD patterns of 1-3.

Figure 6 :

 6 Figure 6:Normalized solid state luminescence of (excitation λ = 370 nm).

Figure S1 .

 S1 Figure 7: Fluorescence decay profiles.

Figure S2 .Figure S3 .Fluorescence

 S2S3 Figure S2. Simulated and experimental PXRD pattern of 2.

  Structural informationcan also be obtained from the excited state lifetime since interactions with neighbouring molecules in the solid state can stabilize or destabilize the excited state. The luminescence profiles of CDPA in CH 3 CN solution, powders of CDPA and CDPA-

				6
	dramatic effect on the supramolecular interactions formed, from usual carboxylic acid-
	pyridine interactions to deprotonation of CDPA. The usual pyridine-carboxylic acid
	intermolecular interactions in 1 blue shift the CDPA solid state luminescence from 587
	nm to 548 nm. Unexpectedly 3-picoline, the weakest base of the picolines used,
	deprotonates the carboxylic acid of CDPA in the solid state. Thus, supramolecular
	interactions have successfully utilized to tune the CDPA solid state luminescence
	gradually from 587 nm to 548 nm.
	F o
	r
	P
		e e r
	Compound CDPA-CH 3 CN CDPA-solid	τ τ τ τ 1 (ns) 1.733874 9.56807	τ τ τ τ 2 (ns) 2.972946 R e v i e w	χ χ χ χ 2 1.004136 1.038992
	CDPA-pyridine	8.292209	1.887961	1.00473
	1	10.58169	2.467983	1.040812
	2	2.369388	10.02734	1.042162
	3	2.621565	7.800654	1.050449

pyridine are shown in figure

7

. Decays monitored in CH 3 CN solution could be fitted monoexponentially, indicating a singlet excited state was exclusively formed. CDPA solids, CDPA-pyridine, and 1-3 showed biexponential decay, suggesting the involvement of multiple excited states and all had exhibited similar lifetimes (Table

-1)

.

Table

1

: Fluorescence lifetimes (τ F ) (Decay monitored at the corresponding λ max ; Excitation at 370 nm).

Conclusion

CDPA forms unusual intermolecular interactions with pyridines in the solid state.

The position of methyl group on the pyridine (4-picoline, 2-picoline, 3-picoline) has a

http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry

Acknowledgement

SPA acknowledges the financial support of EU FP6 [MKTD- CT-2004-014472] and SV the support of Science Foundation Ireland [05PICAI819] in the generation of this work.


Santiago Alvarez
email: santiago@qi.ub.es

Jorge Echeverría

Stephen Hales

Vegetable Staticks

New Perspectives on Polyhedral Molecules and their Crystal Structures

Keywords: continuous shape measures, stereochemistry, shape maps, polyhedranes Continuous shape measures, stereochemistry, polyhedral molecules, polyhedranes, polyhedrenes

Relevant families of ideal polyhedra (Platonic, Archimedean, prisms Johnson, and Fullerenes) are briefly summarized, and an overview of polyhedral alkanes and alkenes, existing or hypothetical, is presented. The assignment of a polyhedral shape to a specific compound with the help of continuous shape measures and derived tools is also briefly discussed, and application of shape analysis to cyclic molecules such as cyclobutane, cyclohexane and cyclooctatetraene is presented to illustrate the usefulness of ideal polyhedra in the stereochemical description of non-polyhedral molecules. Finally, the presence of latent octahedral symmetry in icosahedral polyhedra is used to design new molecules with nested shells of the two supposedly incompatible symmetries, and to explain the cubic crystal structures of icosahedral molecules such as dodecahedrane and Buckminsterfullerene.

Introduction

While the Platonic tetrahedral coordination around a carbon atom, proposed in 1874 by Jacobus Henricus Van 't Hoff and Joseph Achille Le Bel, represented the cornerstone of a new discipline, stereochemistry, molecules with polyhedral skeletons in which a carbon atom occupies each vertex, are relatively newcomers to the world of synthesized and well characterized molecules. The oldest member of the family of polyhedranes, cubane, was reported in 1964. [1] A subsequent geometrical analysis of prisms, Platonic and Archimedean polyhedra, pointed to those that could be made of sp 3 -hybridized carbon atoms. [2] Tetrahedrane had to wait more than a decade to see the light, [3] and soon after dodecahedrane could be synthesized. [4] Much more recent is the discovery of buckminsterfullerene, of formula C 60 , with the shape of a truncated icosahedron. [5] Besides the Platonic solids, there are other sorts of semiregular polyhedra that can be made as purely organic molecules. These include Archimedean solids, prisms and some of the 92 Johnson polyhedra. [6] With the structure of a polyhedral molecule at hand, we must address two relevant stereochemical questions: Which ideal polyhedron represents best its stereochemistry? How similar is the molecular structure to the ideal shape? Avnir and coworkers have proposed that symmetry [7] and shape [8] should be treated as continuous properties and defined continuous symmetry measures (CSM) and continuous shape measures (CShM). These parameters allow us to calibrate the deviation of structures from a given symmetry or shape at the same scale, independent of their size or number of vertices. Later on, we showed that one can define a minimal distortion interconversion path between two polyhedra in terms of continuous shape measures and therefore numerically evaluate not only the deviation of a given structure from a particular polyhedron, but also its deviation from the minimal distortion path between two reference polyhedra. [9] Herein we wish to present an overview of the main polyhedral organic molecules. We will also present examples of minimal distortion paths between a polygon and a polyhedron to organize structural and conformational diversity of cyclohexyl and cyclooctatetraene derivatives. Finally, we will discuss the latency of cubic symmetry in icosahedral and 3 tetrahedral molecules, and on the implications for crystal packing and for the design of metalorganic frameworks.

Shape and Symmetry of Ideal Polyhedra

Although molecular shape and symmetry are intimately associated, it is important to stress here the main differences between these two properties. Let us consider as an example the two Archimedean polyhedra with 24 vertices shown in Figure 1, the truncated cube and the truncated octahedron: they both have octahedral symmetry (i.e., they belong to the O h symmetry point group), but they differ in their number and type of faces. In other words, they have the same symmetry but different shapes.

In brief, we say that two objects (molecules) have the same shape if they differ only in size, position or orientation in space. Alternatively, we can say that two objects (molecules) have the same shape if they can be superimposed by combinations of translations, rotations and isotropic scaling. [10] On the other hand, two objects have the same symmetry if they remain indistinguishable after application of the same set of symmetry operations. It follows that two objects with different shapes may have the same symmetry, and we can therefore conclude that shape is a more stringent criterion than symmetry, as illustrated by the examples in Figure 1. There are, however some cases in which shape and symmetry are equivalent, and these correspond precisely to the Platonic polyhedra, since, e. g., all four vertex polyhedra with tetrahedral symmetry have the same shape, and the same happens for the octahedron, the cube, the dodecahedron and the icosahedron. Among the Archimedean polyhedra, only for the cuboctahedron and the icosidodecahedron are shape and symmetry equivalent. The set of Platonic solids (tetrahedron, octahedron, cube, icosahedron and dodecahedron) are the most regular polyhedra, each having all its edges, faces and vertices equivalent. Second to the Platonic solids in regularity come the Archimedean polyhedra, in which all vertices (but not faces or edges) are equivalent. However, these two families provide us with only a limited number of ideal shapes and we should have at hand other sets of less regular polyhedra. For instance, the prisms, whose ideal shapes are conventionally those with all edges of the same length. As a result, all the faces of the ideal prisms are regular polygons. If more reference shapes are needed we can make recourse to the 92 Johnson polyhedra, [11] defined as those that have as faces only regular polygons with edges of the same length, excluding the Platonic, Archimedean, prismatic and antiprismatic polyhedra. Yet another important family of polyhedra is that of the Fullerenes. Even if the name comes from C 60 , Buckminsterfullerene, that has the shape of the Archimedean truncated icosahedron, it refers to all those polyhedra with twelve pentagonal and any number of hexagonal faces, [12] in which all vertices are threeconnected.

We note that the Platonic and Archimedean polyhedra belong to one of three highsymmetry point groups, icosahedral (I h ), octahedral (O h ) or tetrahedral (T d ), or to their rotational subgroups. In principle, icosahedral and octahedral symmetries are incompatible, [13] since the former features five-fold rotational axes, while the latter has four-fold axes instead. However, we have recently shown [14] that icosahedral polyhedra have latent octahedral symmetry [15] that can be revealed in chemical structures by an appropriate substitution pattern. We will go back to some chemical implications of this symmetry paradox in a later section.

Continuous Shape and Symmetry Measures

According to the proposal of Avnir and coworkers, [7,[START_REF] Avnir | Concepts in Chemistry: A Contemporary Challenge[END_REF] in order to obtain a shape measure for a structure X (represented in 1 by the circles joined by dashed lines) we need first to search for the ideal shape A (represented in 1 by a square) that is closest to our problem structure. This search requires optimization with respect to size, orientation in space and pairing of vertices of the two structures. Once the reference shape is found, we calculate the distances between the equivalent atomic positions in the two structures, q k , from which we calculate the shape measure according to equation 1, where N is a normalization factor that makes the continuous shape measures (CShM) values size independent. To optimize A, S X (A) must be minimized with respect to size, orientation and vertex pairing.

1 S X (A) = min q k 2 k =1 N ! N 100 [1] F o r P e e r R e v i e

w

From the definition of equation 1 it can be shown that the S X (A) values must lie between 0 and 100. The resulting value is zero if the problem structure X has exactly the desired shape A, and will increase with the degree of distortion. As a rule of thumb we can say that chemically significant distortions should give CShM values higher than about 0.1, while values of the order of 2 or higher indicate important distortions. [START_REF] Zabrodsky | [END_REF] Since all CShM values are in the same scale, independently of the reference shape adopted and the number of vertices, we can compare, for instance, the deviation of a given structure from different reference shapes or of different structures with respect to the same ideal shape.

Some Polyhedral Alkanes

Since the fullerenes form a wide family and a huge number of publications have been devoted to them, we concentrate here on the more sparse examples of polyhedral molecules from other families. Some structurally characterized polyhedral alkanes are summarized in Table 1, most of them with Platonic or Archimedean polyhedral shapes. The Johnson and Catalan polyhedra all have vertices with connectivity four or higher, making them poorly adapted for cage alkanes. Nevertheless, if we allow some of the edges of a polyhedron to correspond to non-bonded atoms, accepting some deviation from the ideal polyhedron, we may find a couple of Johnson polyhedranes. One of them is bicyclo(1.1.1)pentane, which resembles a trigonal bipyramid, with structures of its derivatives presenting shape measures relative to such a polyhedron between 1 and 2. Cunneanes (Figure 2), in contrast, deviate significantly from the ideal gyrobifastigium (shape measures of about 8.8), due to the presence of two non-bonded edges. However, the gyrobifastigium is still the polyhedral shape that best describes the structures of cunneanes. Another interesting irregular polyhedron is that showcased by octahedrane (Figure 2), sometimes named "melancholyhedron" because it first appeared in an well known engraving from Albrecht Dürer titled Melancholy. [6] Trigonal Bipyramid Gyrobifastigium Melancholyhedron

Experimental Structures

Tetrahedrane 4 Tetrahedron Platonic [18] Bicyclo(1.1.1)pentane 5 ~ Trigonal Bipyramid Johnson [19] Prismane 6 Trigonal Prism Prism [20] Cubane 8 Cube Platonic [21] Cunneane 8 ~ Gyrobifastigium Johnson [22] Pentaprismane 10 Pentagonal Prism Prism [23] Hexaprismane 12 Hexagonal Prism Prisms [24] Octahedrane 12 Melancholyhedron Irregular [25] Dodecahedrane 20 Dodecahedron Platonic [26] Theoretical Structures Truncoctahedrane 24 Truncated Octahedron Archimedean [27] Fullerane 60 Truncated Icosahedron Archimedean [28] For a polyhedron whose vertices are made of CH groups to be feasible, it must have only three-connected vertices. In the corresponding polyhedrane, the three edges meeting at each vertex corespond to C-C bonds. There are three Platonic and seven Archimedean polyhedra that comply with this requirement, two of which have been reported only as theoretical molecules (Table 1), although they appear scattered in the literature. To have a full gallery of all possible Platonic and Archimedean polyhedranes, we have optimized them via DFT calculations (Figure 3) and found them to correspond to minima in the corresponding potential energy surfaces, with carbon-carbon bond distances characteristic of single bonds. It is customary to relate the energies of polyhedral molecules to the deviation of the carbon atoms from the tetrahedral geometry imposed by the shape of the polyhedron. Thus, a strain energy is defined, which is calculated as the total energy divided by the number of CH groups, taking as zero strain energy the value obtained for the most tetrahedral case, which corresponds to dodecahedrane (equation 2). The resulting strain energies show the expected dependence on the H-C-C bond angles, and the most stable polyhedra are those that present average H-C-C bond angles closer to the tetrahedral angle ( The geometries of some Archimedean polyhedra are not well suited for fully dehydrogenated C n molecules, because of the non-planarity of their vertices (measured by the sum of the subtended C-C-C bond angles, Σ, that should be 360º for an ideal sp 2 carbon atom, see Table 2). Nevertheless, small deviations from planarity can be withstood, as in the truncated icosahedron of C 60 . Theoretically optimized structues of other Archimedean polyhedrenes have been reported in separate studies (Table 3), to which we can add the truncated cube. It must be noted that different delocalization schemes may appear in these molecules, as evidenced for several alternative structures of the Platonic dodecahedrene C 20 . [29]

Shape Maps and Interconversion Pathways

For the stereochemical analysis of families of compounds we have found it useful to represent scatterplots of their shape measures relative to two alternative ideal polyhedra with the same number of vertices (e.g., A and B), that we call shape maps. In these maps, the lower left limit always corresponds to the interconversion path between the two reference shapes, usually polyhedra or polygons. The shape measures of all structures i along the minimal distortion interconversion path between polyhedra A and B, S i (A) and S i (B), must obey the following relationship: [9] arcsin S i (A) 10 + arcsin S i (B) 10 = ! AB [3] where θ AB is a constant for each pair of polyhedra, the symmetry angle. [9] Structures that do not belong to the minimal distortion path do not obey equation 3 and their distance to that path can be calibrated by means of the path deviation function defined in equation 4, where x refers to an arbitrary structure.

! x A, B ( ) = 1 " AB arcsin S x (A) 10 + arcsin S x (B) 10 # $ % % & ' ( ( ) 1 [4]
Furthermore, for structures that are along the minimal interconversion pathway we have defined a generalized interconversion coordinate, [35] that measures the percentage of the path between two polyhedra covered by the problem structure (equation 5).

! A" B = 100 # AB arcsin S i (B) 10 $ % & ' ( ) [5]
With such tools we can (a) detect very easily those structures that are intermediate between two ideal shapes, (b) obtain a quantitative description of how close (or how far) a given structure is from a path, (c) obtain molecular models of the shapes that correspond to steps along the interconversion path, and (d) calibrate the distance of the problem structure to the two extremes of the path. According to the proposal that crystal structure data offer clues to reaction pathways, [36] the analysis of crystal structures from the point of view of minimal distortion paths should be helpful in gaining insight into chemical reactivity aspects.

To illustrate the possible applications of shape maps, we show in Figure 4 the structural data of all non-fused cyclohexyl groups found in the Cambridge Database in two different shape maps. In the first one, the structures are compared with the planar hexagon and the octahedron, since its chair conformation can be seen as an intermediate between those two ideal shapes (Scheme I). We have found that 97% of the cyclohexyl groups found are aligned along the minimal distortion path between the hexagon and the octahedron, with a deviation of at most 10% (the small portion of molecules that deviate most from that path are not shown in the plot for clarity).

An alternative way of looking at the data plotted in Figure 4 consists in analyzing the frequency of structures found at different steps along the path, shown in the histogram of Figure 5. There we see that the structures are strongly concentrated around a generalized coordinate ϕ OC→HP of 74%, suggesting that such a conformation is the most stable one. In effect, a DFT optimization of the molecular structure of cyclohexane in the chair conformation appears at a generalized coordinate ϕ OC→HP of 74.8%, exactly on the track from the octahedron to the planar hexagon. We figured out that the cyclohexyl groups that significantly deviate from this path should probably present a boat conformation. Therefore, we have plotted those structures in a shape map relative to the hexagon and the trigonal prism, since the boat structures are in-between those two ideal shapes. The results (Figure 4, right) clearly discriminate the boat from the chair structures, but also allow us to detect different degrees of each of those two conformations. Of course, it is not possible to ascribe in an unequivocal way a boat or chair conformation to those molecules that are close to the planar hexagon.

Scheme II

A second case that we have analyzed is that of the cyclooctatetraene groups (COT). Those skeletons are expected to be in a boat shape (reminiscent of the cube) for the neutral COT, but perfectly octagonal for the aromatic dianion. It is not surprising, therefore, that the structures of all C 8 R 8 groups found in the CSD are aligned to a good approximation along the coresponding minimal distortion interconversion path (Figure 6). It is interesting to note that there is no geometric discontinuity between the aromatic dianionic rings and the neutral tetraenes. In addition, the absence of structures in a wide portion of the pathway is suggestive of a high energy barrier for the interconversion of the two isomers, cyclooctatetraene and cubane, through the pathway analyzed here. Let us finally stress that cyclooctatetraene, C 8 H 8 , is not among the systems that most deviate from planarity. Some of the compounds that appear along the path are shown in Scheme II, together with their shape measures.

Finally, a look at the structures of cyclobutane derivatives (disregarding those with fused rings) shows also that their conformations can be nicely described as being along the square to tetrahedron pathway (Scheme III). Among 551 crystallographically independent data sets found, all cyclobutane skeletons deviate less than 5% from that path with only two exceptions, covering the range from strictly planar to 28% bending toward the tetrahedron.

Scheme III

Square Tetrahedron

Symmetry Paradox

We have recently shown that octahedral symmetry is latent in polyhedra of icosahedral symmetry, [14] even if the corresponding symmetry point groups are incompatible. A consequence of such a relationship is the capability of molecules with icosahedral symmetry to form a network of intermolecular interactions with octahedral symmetry that results quite often 4 we present some examples of organic polyhedral molecules that form cubic crystals. Compd.

Packing [a] Polyhedron Ref.

C 20 H 20 (dodecahedrane) fcc Dodecahedron [37] C 60 (fullerene) fcc Truncated Icosahedron [38] C 60 •H 2 C=CH 2 fcc Truncated Icosahedron [39] K 3 Ba 3 C 60 bcc Truncated Icosahedron [40] C 60 •O 2 fcc Truncated Icosahedron [41] [a] fcc = face-centered cubic; bcc = body-centered cubic.

We think that octahedrally-arranged atoms or functional groups can also be attached to icosahedral molecules. We have noticed, for instance, that buckminsterfullerene can be considered as formed by six fulvalene units whose centers are arranged in an octahedral way. Alternatively, it can also be described as an assembly of six naphthalene units, whose centers occupy the vertices of an octahedron (Figure 7). Chemical substitutions that occupy the centers of those bonds would result in an octahedron circumscribed around the C 60 truncated icosahedron. As an example, we have computationally optimized the structure of a hexaepoxidized fullerene, in which the six oxygen atoms are seen to form a perfect octahedron (Figure 8). In a related experimental structure, [42] the same positions are occupied by cyclopropane rings, whose carbon atoms also form a perfect octahedron. In both cases, the six substituents directly attached to the fullerene have octahedral shape measures smaller than 0.01, while the fullerene core deviates very little from its truncated icosahedral shape (shape measures of 0.03 and 0.06 for the epoxidized and cyclopropanated derivatives, respectively). ). Right: Analogous octahedron formed by six cyclopropane fused rings in an experimentally reported compound. [42] Another computational example of a molecule that combines the icosahedral symmetry of a dodecahedrane skeleton with the cubic symmetry of eight substituents is that of C 20 H 12 Br 8 , shown in Figure 9. Although adamantane is not usually perceived as a polyhedral molecule, it can be described as a composite of a tetrahedron of tertiary carbon atoms (2) bridged by secondary carbon atoms that form a circumscribed octahedron. [6] Omar Yaghi and coworkers [43] have taken advantage of the tetrahedron implicit in adamantane to design a network of a porous metal organic framework (MOF) reminiscent of the PtS structure but with larger voids. The tetrahedral adamanane unit is provided with four carboxylate groups that have the same spatial connctivity than a sulfide ion. Those carboxylates are coordinated to a pair of copper(II) ions

Concluding Remarks

An assortment of beautifully symmetrical polyhedra can be expressed as organic molecules, and several examples have been obtained in the bench or in the computer. We have shown here also that thinking in terms of polyhedral shapes may be helpful to analyze the stereochemistry of non polyhedral molecules. The implicit polyhedra found in adamantane, for instance, has beeen used by Yagi and coworkers to design and build up porous extended frameworks. The latent cubic symmetry present in molecues with icosahedral symetry is important in establishing the directions of intermolecular interactions that often result in crystals with cubic packings, as in dodecahedrane and buckminsterfullerene. DFT calculations were carried out with Gaussian03, [44] using the B3LYP hybrid functional and a 6-31G** Gaussian basis set. [45] All calculated structures reported were characterized as true minima through vibrational analysis. Structural searches were carried out in the CSD, [46] version 5.30 with three updates, and the atomic coordinates transferred to the SHAPE code [47] to calculate continuous shape measures, path deviation functions and generalized coordinates. [48]

Figure 1 .

 1 Figure 1. Two Archimedean polyhedra with octahedral symmetry and 24 vertices: the truncated cube (left) and the truncated octahedron (right).

Figure 2 .

 2 Figure 2. Structures of the carbon skeletons of bicyclopentane, cunneane and octahedrane, that approximate the geometry of the trigonal bipyramid, the gyrobifastigium, and the melancholyhedron, respectively.

Figure 3 .

 3 Figure 3. Calculated structures of the Archimedean alkanes.

Figure 4 .

 4 Scheme I

11 Figure 5 .

 115 Figure 5. Frequency of cyclohexyl groups as a function of their degree of hexagonality (generalized coordinate for the octahedron-planar hexagon conversion), in a logarithmic scale.

Figure 6 .

 6 Figure 6. Structural data of cyclooctane, cyclooctatetraene (COT) and cubane derivatives, plotted in a shape map relative to the cube (CU-8) and the planar octagon (OP-8). The line represents the minimum distortion interconversion path between the octagon and the cube.

  structures of icosahedrally symmetric molecules. In Table

Figure 7 .

 7 Figure 7. Buckminsterfullerene C 60 as an assembly of six naphthalene units (left) or six fulvalene units (right).

Figure 8 .

 8 Figure 8. Left: Epoxidized fullerene whose six oxygen atoms form a perfect octahedron, templated by the C 60 truncated icosahedron (computational DFT results). Right: Analogous octahedron formed by six cyclopropane fused rings in an experimentally reported compound.[42]

Figure 9 .

 9 Figure 9. Optimized structure of an octabrominated dodecahedrane C 20 H 12 Br 8 with a substitution pattern that displays a cubic set of bromine atoms (hydrogen atoms not shown for clarity) circumscribed around the dodecahedron of carbon atoms.

  four-connected unit(3), that plays the same role of the Pt II ion in PtS. The analogous construction principle of the two structures can be appreciated in Figure10.

2 3 Figure 10 .

 2310 Figure 10. Formal replacement of the tetrahedral sulfide ions in PtS by adamantane units (2) and of the square planar Pt II ions by Cu 2 (carboxylate) 4 building blocks (3).

Table 1 .

 1 Structurally characterized polyhedranes.

	Compound	n	Polyhedron	Family	Ref.

Table 2

 2 ).

	E s (C n H n polyhedrane) =	E total (C n H n polyhedrane) n	!	E total (C 20 H 20 dodecahedrane) 20	[2]

Table 2 .

 2 Strain energies (E s , kcal/mol) and bonding parameters for several optimized polyhedranes with general formula C n H n . All distances in Å, angles in degrees; experimental values given in parentheses when available.

	n	Polyhedron	E s	C-C (Å)	H-C-C (º)	Σ (º) [a]
	120 Trunc. Icosidodecahedron	23.4	1.58-1.61	101.7	354
					96.7
	60 Truncated Dodecahedron	25.4	1.545	101.7	348
				1.534	97.5
	60 Truncated Icosahedron	7.4	1.571	101.6	345
				1.557
	48 Truncated Cuboctahedron	13.0	1.579	102.8	345
				1.546
	24 Truncated Cube	13.2	1.525	108.0	330
				1.513
	24 Truncated Octahedron	5.8	1.569	109.2	330
				1.533	107.2
	20 Dodecahedron	0.0	1.556 (1.545)	110.9 (111)	324
	12 Truncated Tetrahedron	6.7	1.522	116.1	300
				1.499	114.2
	12 Hexagonal prism	11.3	1.560	115.1	300
		(hexaprismane)		1.566	122.1
	10 Pentagonal prism	10.7	1.561 (1.55)	119.4 (119)	288
		(pentaprismane)		1.570 (1.57)	123.4 (124)
	8 Cube	16.7	1.571 (1.551)	125.3 (126)	270
	6 Trigonal Prism (prismane)	20.8	1.522 (1.53)	129.6 (130)	240
				1.558 (1.55)	132.7 (132)
	4 Tetrahedron	31.1	1.479 (1.486)	144.7 (145)	180
	[a] Sum of C-C-C bond angles around a carbon atom.

Table 3 .

 3 Reported theoretical studies of Archimedean and Platonic polyhedrenes with general formula C n .

	n	Polyhedron

Table 4 .

 4 Cubic crystal structures of molecules with icosahedrally symmetric carbon polyhedra.

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Acknowledgments

This work has been supported by the Ministerio de Invesigación, Ciencia e Innovación (MICINN), project CTQ2008-06670-C02-01-BQU, and by Generalitat de Catalunya, grants 2009SGR-1459 and XRQTC. Allocation of computer time at the Centre de Supercomputació de Catalunya, CESCA, is greatfully acknowledged. J. E. thanks the Spanish Ministerio de Educación for an FPU grant (reference AP2008-02735).