Bivariate normal distribution

Bivariate normal distribution

In a previous blog post, I have discussed the case of normally distributed random variables. But what happens if we have a vector whose components are random variables with this kind of distribution? In particular, what is the expression of the joint probability density? I don’t know how to derive this formula from the marginal distributions, but I have been able to demonstrate, after several hours of work, the following proposition.

Proposition 1. The joint probability density in the case of a random vector whose two components follow a normal distribution is:

probability density.png

where σ_1 and σ_2 are the standard deviations, μ_1 and μ_2 are the expectations of the two random variables and ρ is the correlation coefficient between the two variables.

Proof. The marginal distribution of x_1 is given by

marginal density.png

By operating the substitutions

substitutions.pngwe obtain

marginal density 2.png

This demonstrates that x_1 is in fact a normally distributed random variable and that its expectation and standard deviation are σ_1 and μ_1, respectively. The same applies to x_2. In order to demonstrate that ρ is the correlation coefficient between x_1 and x_2 we have to calculate the covariance of f_X, which is given by E[X_1,X_2]-E[X_1]E[X_2]. The first addend is given by:

covariance 1.png

Let’s consider the inner integral and the following substitutions

substitutions 2.png

We have:covariance 2.png

The two integrals inside the root square are not the easiest ones, and I will not discuss them here. Just consider that they can be solved using the well known results:

exponential integral 4

We have exponential integral 3.png

 If we put these two integrals inside I, we obtain

covariance 3.png

The two inner integrals are two difficult ones, again. The first one can be solved using the same exponential integral used above:

covariance 4

For the other inner integral we have

covariance 5.png

The first and the second integrals can be solved using the table previously reported, while the third one is zero. At the end of all this work, we have:

covariance 6.png

And this proves that ρ is the correlation coefficient between x_1 and x_2. This is a complete demonstration of the expression of the joint probability density of a bivariate normal distribution ■

This formula can be extended to the case of m components.

Proposition 2. The joint probability density in the case of a random vector whose m components follow a normal distribution is:

probability density 2.png

where C is the covariance matrix, given by

covariance matrix.png

Proof. This formula is easy to prove in the case of m=2, using proposition 1. I have no idea about how it could be proved when m>2 ■

I have written a code in Fortran (download) that asks you for expectations, standard deviations and correlation coefficient and plots the joint density of the bivariate normal distribution. I have written a code in Octave that does the same job (download). In the picture below you find the joint density for two standard normally distributed random variables with ρ=0 (left), ρ=0.7 (right, upper half) and ρ=-0.7 (right, lower half). These pictures have been plotted by the code in Fortran. The other image is a comparison between the two programs, for another bivariate normal distribution.






End ME/CFS Worldwide Tour (Italy)

End ME/CFS Worldwide Tour (Italy)

These are the slides that I used in my presentation during the Italian stage of the End ME/CFS Worldwide tour.

I have not thanked enough Chiara Sacchetto, Girolamo Carollo, Giada Da Ros, Valentina Viganò, Maria Pia Cavalet and Angela Colucci for their hard work in organizing this meeting. And of course, my English is probably not good enough to express my gratitude to Linda Tannenbaum and the Open Medicine Foundation. I have no words for that.

There are a few slides in Italian, the others are in English. I have included also some slides that I didn’t have the chance to show during the meeting. Download the slides.


Three new possible autoepitopes in ME/CFS

Paolo Maccallini


I have performed a set of analysis of experimental data already published about autoimmunity to muscarinic receptors in ME/CFS. My predictions are that extracellular loop 2 and 3, and also transmembrane helix 5 of both muscarinic cholinergic receptors 4 and 3, are main autoantigens in a subset of ME/CFS patients. Moreover, I have found that autoimmunity to M4 and M3 ChR is independent from autoimmunity to beta 2 adrenergic receptor, also reported in ME/CFS patients.  


Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a debilitating disease characterized by cognitive deficits, fatigue, orthostatic intolerance with symptoms exacerbated after exertion (post-exertional malaise, PEM) (IOM, 2015). This disease has no known cause but several abnormalities have been observed in energy metabolism (Tomas C. and Newton J. 2018), immune system, gut flora (Blomberg J. et al. 2018), brain (Zeineh MM. et al. 2014). A possible role for autoantibodies in the pathogenesis of the disease has been suggested by the finding of reactivity of patient sera to two nuclear antigens (Nishikai, et al., 1997), (Nishikai, et al., 2001), to cardiolipin (Hokama, et al., 2009), to HSP60 (Elfaitouri A. et al. 2013), and to muscarinic cholinergic (M ChR) and beta adrenergic receptors (ß AdR) (Tanaka S et al. 2003), (Loebel M et al. 2016); reactivity that was significantly elevated when compared to healthy contols. Reactivity to adrenergic and muscarinic Ch receptors has been confirmed by two independent groups, but these results have not been published yet (R). A role for autoantibodies in at least a subgroup of patients has also been suggested by a response to rituximab, a CD20 B cells depleting agent (Fluge Ø. et al. 2011), (Fluge Ø. et al. 20115), and to immunoadsorption (Scheibenbogen C. et al 2018). Sera response to muscarinic cholinergic receptors is confirmed in two studies but both of them used an immune assay with proteins coated on a plate. This kind of test does not allow to identify the exact autoepitope on the receptor and – even more importantly – it is subjected to false positive results because it exposes to sera surfaces of receptors that are hidden when they are in their physiological position (Ramanathan S et al 2016). Nevertheless, the amount of data provided in the study by Loebel et al. where reactivity of sera to 5 subtypes of muscarinic cholinergic receptors have been measured simultaneously, has – in our opinion – the potential to unveil the exact autoepitope(s). Thus, I performed a bioinformatical analysis on experimental data from this study in order to extract hidden information. I used a software for the in silico study of B cell epitope cross-reactivity (Maccallini P. et al. 2018) and a software for amino acid protrusion index calculation (Ponomarenko J. et al., 2008).  Our prediction is that patients sera mainly react to three epitopes that belong to the second and third extracellular loop of M3 and M4 ChR, but also to a hidden epitope of the same two receptors, leading to possible false positive results of this test. I have also found that the reactivity to beta 2 adrenergic receptor (ß2 AdR) found in the study by Loebel et al. is not due to the same antibody that reacts to muscarinic cholinergic receptors.


Search for cross-reactive epitopes. Cross-reactivity between muscarinic cholinergic receptors M4 and M3, and between M4 and M1 has been studied in silico using EPITOPE, a software already described (Maccallini P. et al. 2018). Briefly, EPITOPE searches for cross-reactive epitopes shared between two proteins (let’s say protein A and protein B) by comparing each possible 7-mer peptide of A with each possible 7-mer peptide of B. The comparison is made using the algorithm by Needleman and Wunsch (Needleman SB. and Wunsch CD. 1970)  with a gap model a + b·x, where a is the opening gap penalty, b is the extending one, and x is the extension of the gap. A penalty for gaps at the end of the alignment was also assumed. The choice for gap penalties and substitution matrix were done according to the theory already developed for peptide alignments (Altschul SF. 1991), (Karlin S. and Altschul SF. 1990). Available experimental data on cross-reactivity between γ enolase and α enolase (McAleese SM. et al. 1988)  have been used for EPITOPE calibration: a score >60 was considered the cut-off for cross-reactivity, a score below 50 indicates non-cross-reactive epitopes; a score between 50 and 60 defines a borderline result. A simpler version of EPITOPE has been used for single local alignments. The main program used for M4-M3 comparison, its subroutine NeWalign and the substitution matrix employed are available for download. Primary structures used in this work have been downloaded from UniProt and are the following ones: M1 ChR (P11229), M3 ChR (P20309), M4 ChR (P08173), B2 AdR (P07550).

Surface exposure. In order to select only those 7-mer peptides that are on the surface of proteins, I have considered their mean protrusion indexes. A protrusion index of at least 0.5 has been considered the cut-off for surface exposure. Protrusion indexes of single amino acids have been calculated with ElliPro. A protrusion index of 0.5 means that the amino acid is outside the ellipsoid of inertia which includes 50% of the centers of mass of all the amino acids of the protein (Ponomarenko J. et al., 2008). For M4 ChR I have used the crystal structure 5DSG (Thal DM. et al. 2016). The 3D structure of human M3 ChR has not been experimentally determined yet, so I have used a theoretical model built using murine M3 ChR (PDB ID: 4DAJA) as a template, provided by ModBase.

M ChR plot
Figure 1. The position of the first amino acid of each possible 7-mer peptide of M4 ChR is reported on the abscissa, the score for the comparison of each of these peptides with M1 ChR (blue line) and M3 ChR (orange line) is reported on the ordinate. N terminus, extracellular loop 1, 2 and 3 are also indicated. Scores above the yellow line indicate cross-reactivity, scores below the blue line indicate a lack of cross-reactivity.

Selection criteria. Our purpose is to predict whih epitopes of M3 and M4 ChRs sera from ME/CFS patients react to. So, we search for M4 ChRs 7-mer peptides that are cross-reactive to M3 ChR, but non-cross-reactive to M1 ChR. Moreover, they must present surface exposure both on M4 and on M3 ChR (otherwise antibodies can’t reach them). So, selection criteria for M4 ChR epitopes are as follows:

  1. they must be cross-reactive to M3 ChR;
  2. they must be non-cross reactive to M1 ChR or borderline;
  3. they must present a mean protrusion index ≥0.5;
  4. M3 ChR peptides they cross-react to have to present a mean protrusion index ≥0.5.

We will refer to strict criteria when we assume only non-cross-reactivity in 2, while weak selection criteria are fulfilled when M4 ChR epitopes have borderline reactivity to M3 Chr peptides.

M4 vs M1, M3
Figure 2. Distribution of the scores from the comparison of M4 ChR with M1 ChR (left) and with M3 ChR (right). M3 ChR presents a slightly higher mean score.


The search for 7-mer peptides of M4 ChR that are cross-reactive to M3 ChR found 108 sequences. We then studied cross-reactivity to M1 ChR for each of these peptides and we found that 11 of them are non-cross-reactive and that other 9  peptides have borderline reactivity. None of these 20 peptides presented a cross-reactivity to B2 AdR (Table 1S, column 1). Scores between peptides of M4 ChR and the other two muscarinic cholinergic receptors are plotted in Figure 1. The distribution of scores from the comparison of M3 ChR with M1 ChR and with M3 ChR are reported in Figure 2. For the M4 ChR 20 epitopes mentioned above, we calculated the mean protrusion indexes and we did the same calculation for their cross-reactive peptides on M3 ChR. We also indicated their position with respect to the plasma membrane. All these data are collected in Table 2S. Once we apply selection criteria on these 20 peptides, we obtain 9 epitopes (Table 1). Of these selected epitopes, one belongs to a transmembrane helix: peptide 186-192 of M4 ChR, which cross-reacts to peptide 231-237 of M3 ChR. Peptide 418-431 of M4 ChR is partially immersed in the plasmatic membrane, even though its cross-reactive peptide of M3 ChR is entirely exposed to the extracellular space, and the same applies to the other two epitopes found (figure 1). Peptide 175-181 of M4 ChR cross-reacts to peptide 211-217 of M3 ChR; peptide 186-192 of M4 Chr cross-reacts to peptide 222-228 of M3 ChR; peptide 418-431 of M4 Chr cross-reacts to peptide 513-522 of M3 ChR. Sequences that fulfill selection criteria and their respective inverted sequences are collected in  Table 2.

Table 1
Table 1. This is the collection of M4 Chr 7-mer peptides that are cross-reactive to M3 ChR; are not cross-reactive or borderline with M1 ChR; have a mean protrusion index higher than 0.5; are cross-reactive with epitopes of M3 ChR with a protrusion index higher than 0.5.


B cells autoimmunity to muscarinic cholinergic receptors in ME/CFS has been reported in two studies (Tanaka S et al. 2003), (Loebel M et al. 2016) and this finding has been recently confirmed by two other independent groups who have not published yet (R). The two studies mentioned used full-length proteins coated on a plate in order to perform the immune assay. With this kind of technique we may have both false positives (due to the fact that sera react with peptides that are not in the extracellular domain) and false negatives (due to protein denaturation, which leads to the formation of epitopes that would not be present if the protein were correctly folded) as it has been reported in the case of anti-MOG antibodies (Ramanathan S et al 2016). A way to solve the possible inaccuracy of these data would thus be to measure sera reactivity with a cell-based assay (CBA) which is a test where receptors are expressed by eukaryotic cells and thus they are held in their physiological position.

Figure 1. Peptides of table 1 that belong to the extracellular domain of M3 and M4 ChR are here highlighted directly on the 3D structures of their respective receptors.

Nevertheless, we can still try to extract hidden information from experimental data and predict the position of the epitope(s) ME/CFS patients sera react to. Knowing that sera from patients react to M4, M3 ChRs and that there is a low correlation between reactivity to M4 ChR and reactivity to M1 ChR (Loebel M et al. 2016), we selected 7-mer peptides of M4 ChR that cross-react (in silico) to M3 ChR but not to M1 ChR (Table 2S). We then selected, among them, only those peptides that have surface exposure on their respective proteins (Table 1). The result is that patient sera react to extracellular loops 2 and 3 of both M3 and M4 ChRs (Figure 1), but also to a hidden antigen, a peptide of transmembrane helix 5 of both M3 and M4 ChR.

Our results are of interest because extracellular loops 2 and 3 of M3 ChR are known autoepitopes in Sjögren’s syndrome (Ss) (Deng C. et al. 2915). Moreover, sera from patients with orthostatic hypotension (OH) react to extracellular loop 2 of M3 ChR, where they show an agonistic effect, thus acting as vasodilators (Li H. et al. 2012). OH, a form of orthostatic intolerance has been reported in ME/CFS patients (Bou-Holaigah et al. 1995) while fatigue similar to post-exertional malaise has been described in Ss (Segal B. et al. 2008). A pathogenic role of these antibodies in fatigue for both ME/CFS and Sjögren syndrome could perhaps be due to their vasodilatory effect.

Our analysis unveiled reactivity to a hidden autoepitope, which belongs to transmembrane helix 5 of M3 and of M4 ChR. This epitope is buried inside the plasma membrane when these two receptors are in their physiological position, so this reactivity can’t contribute to the pathogenesis of ME/CFS.

None of the 7-mer peptides of M4 ChR that cross-react to M3 ChR and at the same time don’t cross-react to M1 ChR presents in silico reactivity to B2 AdR. This means that in those patients whose sera present reactivity to both M4-M3 ChR and B2 AdR, there are two distinct autoantibodies. This prediction of our model is consistent with the low correlation found by Loebel and colleagues between anti-M4 ChR and anti-B2 AdR antibodies (Loebel M et al. 2016).

Most B cells epitopes on non-denaturated proteins (i.e. proteins that conserve their tertiary structure) are believed to be conformational (Morris, 2007), so a significant limitation of this study is due to the fact that our analysis considers only linear epitopes. Nevertheless, the main limitation of this study remains by far my encephalopathy.


This analysis of previously published data suggests a role for the second and the third extracellular loop of M4 and M3 ChR as autoantigens in ME/CFS. It also predicts the presence of a hidden autoantigen and thus a risk of false-positive results with standard ELISA.  The eight peptides found by this analysis and their inverse sequences (Table 2) should be employed as query sequences for the search for possible triggering pathogens and for other autoantigens. These predictions should be tested using both cell-based assays and ELISA tests with these 8 peptides coated on the plate.

Table 2.PNG
Table 2. Peptides belonging to M4 and M4 ChR that fulfill our selection criteria are collected on the left. On the right, their reverse sequences. These 16 peptides can be used in BLAST in order to serach for triggering pathogens and for other possible autoepitopes.

Supplementary material. The following two tables represent the first two steps of the analysis presented in this paper. M4 ChR 7-mer peptides that are cross-reactive to M3 ChR are collected in Table 1S, while those of them that are non-cross-reactive (or borderline) to M1 ChR are collected in Table 2S.

Table 1S. Peptides of M4 ChR that are cross-reactive to M3 ChR are collected in the first column. In the second column are collected the scores of these 7-mer peptides obtained from the comparison with M1 ChR. For those that obtained a score below 60, the score from the comparison with B2 AdR is reported in column 5. Positions of peptides of interest that belong to M3ChR and B2 AdR are collected in columns 4 and 6 respectively.
Table 2S.PNG
Table 2S. These 20 peptides are those M4 ChR peptides that cross-react to M3 ChR and at the same time are non-cross-reactive or borderline when compared to M1 ChR. Reactivity to B2 AdR is also indicated, as well as positions with respect to the plasma membrane and mean protrusion indexes. On the left are indicated those peptides of M4 ChR that pass the selection according to our criteria. Both a strict selection and a selection with more weak criteria are reported.

Paolo è una stella

Di Patrizia Giorgi

Paolo è una Stella. Ho conosciuto Paolo di persona all’Ospedale di Trieste. “C’è una persona nelle prima stanza che dice di conoscerla” – mi ha detto l’infermiera, mentre disfavo la borsa e cercavo di prendere confidenza con l’armadietto e l’aria del reparto.

Paolo è un giovane uomo con una mente che abbaglia, è una Stella. E il mio, non vorrei venire fraintesa, è un affetto tra esseri umani che per motivi legati al non essere nel Mondo a causa della scarsa energia, diventa ancora più importante. Chi mi conosce forse sa che è nella mia natura l’essere curiosa di conoscere le persone, e la particolarità di questa malattia che risucchia tutto – risucchiando l’energia – non ha soffocato questa mia inclinazione, anzi, l’ha acuita. Una malattia questa che cambia la visione delle cose importanti sino a comprimerle all’essenza. Ed eliminare il superfluo diventa così vitale ed essenziale per preservare energia! Ad esempio, io piego i panni nella bacinella per un paio di ore, sovrapponendoli; solo dopo li stendo, così le pieghe piu grandi non ci sono piu e risparmio o mi evito di stirare.

Stavo per affidarmi per la prima volta, quel giorno dell’arrivo in ospedale, a Trieste. Diagnosi da pochi mesi, dopo 14 anni di ipotesi, dopo due anni di analisi varie. Sapere che c’era Paolo ricoverato due stanze prima di me mi ha rassicurato molto. Grazie alla sua presenza ho avuto molta meno paura. E non è poco in quella situazione di dubbi.

Per chi non lo sapesse esistono forum e pagine facebook, per diverse patologie,  di persone che condividono paure, ricerche, emozioni; pagine in cui ci si scambia informazioni e ricerche mediche. Autismo, sclerosi multipla, ME/CFS etc. I malatti rari, sopratutto se fra gli invisibili (sottocategoria dei rari, che già visibili lo sono poco), queste persone che brancolano, hanno da questo scambio di informazioni tratto vantaggi. Chi, per problemi di salute, deve trascorrere in casa la maggior o la totalità dei suoi giorni ha bisogno di scambi, rassicurazioni e conforto che solo un minimo contatto umano puo dare. Diverse persone che anni fà rincontrai su facebook, le conoscevo da anni, dal forum di ZAC (CFSitalia).

Sono trascorsi molti anni da allora, da quando ebbi la prima diagnosi CFS e FM (2007, Ospedale di Chieti) e cercavo di capire di cosa si trattasse. Avevo incominciato a cercare da poco. Le informazioni datemi erano quasi nulla: nessuna terapia proposta e nessuna visita di controllo futura. Ringrazio di cuore Zac d’avere aperto quel forum: grazie a lui ho potuto non sentirmi totalmente smarrita e ho avuto informazioni importanti. Mi hanno tutti aiutato a calmare lo smarrimento e la paura in quegli anni di domande senza risposte e di allettamento totale, in cui lo smarrimento era anche di chi convive con me, incredulo nel vedermi in quelle condizioni e spaventato, perché lasciato solo a occuparsi di me.

Tra di noi si condividono paure e c’è uno scambio emotivo molto forte: pur non essendoci mai visti e mai abbracciati se non virtualmente, si creano legami. Si diventa confidenti. Forse anche perché nei forum in quel periodo c’era l’abitudine di non usare il proprio nome. E si sà cosa si intende con “linea rossa” sul forum e chi è Paul Cheney. Ci sono quelli che usano queste comunità virtuali per cercare di capire cosa possono fare, altri per condividere esperienze; ci sono famigliari di malati che cercano risposte, chi cerca aiuto e chi lo dà. Il forum di Zac mi ha fatto conoscere oltre a lui diverse persone (Marco è un esempio). Su facebook, non ricordo come, diversi anni fà alcuni li ho riconosciuti, per la prima volta, col nome reale. Nel forum si usavano nikname: io ero “vivo”, un diminutivo che usavano quelli dei primi tempi. Il nome completo era “vivolenta” (per ovvi motivi).

Adesso mi viene piu difficile scrivere. Ci vado e leggo, cerco informazioni delle persone con cui parlavo e ritrovo racconti di nuovi iscritti, purtroppo sempre alle prese con gli stessi problemi. Ma direi che la comunità del forum continua il suo lavoro in maniera egregia. più volte Zac pensava di lasciarlo in lettura e anche lui, nonostante la sua grave sintomatologia, ha fatto una gran cosa. E forse ho ringraziato poco pure lui. Quindi Grazie Zac di cuore, anche a te.

E’ stato un bel regalo del destino quella coincidenza in ospedale con Paolo. E siccome sono solita non restare indifferente alle relazioni umane, temo di non averglielo sottolineato abbastanza, a Paolo, quanto sia dispiaciuta per lui, come sono dispiaciuta per tutte le persone conosciute nel forum, che questo stare male abbia spezzato i nostri sogni. Le sue ali Paolo le ha sempre.

Grazie Paolo. Grazie Zac e grazie anche a tutti gli altri. E grazie Marco, giovane uomo dal grande cuore. Grazie Daniela. Tutto un mondo di persone invisibili, perché bloccate nei loro letti e di cui il mondo non puo godere, nè loro di lui, se non in minima parte. Persone di cui nessuno si occupa, se non i famigliari, quando ci sono. Vite troppo essenziali per riuscire a trovarle nel mondo reale.

Comporta, l’essenzialità, un sacco di cose. Mi spiace, non ho energe per spiegare meglio, ma l’intento è ringraziare pubblicamente le persone incontrate. Tante ne ho perse, secondo me anche perchè è diverso avere a che fare con questa condizione sconosciuta e difficile anche solo da far capire (potrei sprecare gli anedotti). Non avendo energie da buttare, passi velocemente al dunque. Per alcuni forse troppo velocemente. Le regole sociali inutili sono le prime a saltare e seguono diverse anomalie.

Anche Paolo ha una visione sua delle cose e le racconta nel suo Blog. Non avete idea di quale mente brillante abbia, ed è una perdita per tutti che non sia attivo quanto lo sarebbe se fosse in piena salute. E’ come potere disporre delle fibre ottiche e non usarle. Paolo è comunque ad alta velocità, rispetto alla media, e spero davvero e di cuore per lui possa questa faccenda diventare esperienza quando, e non se.

Ci si spiega e ci si racconta. Quando non spiego vengo fraintesa e quando spiego, spesso sono non creduta. Se si partisse da quello, l’essere creduti, sarebbe già molto di aiuto. Ma accade raramente. Perché ringrazio? Perché è essenziale farlo anche senza un motivo.

Paolo, scrivendo molto meglio di me, potrà spiegare a quanti vorranno approfondire. Sarà un piacere leggerlo, per quanti vorranno farlo, e sarà un arricchimento. Paolo ha fatto ricerche mediche e pubblicato pure, e consiglio di dare un’occhiata a tutte quelle riflessioni e connessioni e analisi che nel suo blog pubblica e condivide, e anche a parecchi medici consiglierei di dare un’occhiata ai suoi studi. Conoscere è nella nostra natura e lui è una mente sopraffina. Come dispiacersi. E come non dispiacersi. L’indifferenza e l’isolamento sono duri da affrontare, di piu se stai male. E sono tristi, non solo per chi la vive, ma anche per chi perda l’occasione di conoscere e godere della nostra compagnia e non ultimo per il contributo che la società perde dalla nostra esclusione.