Ronald Davis at Columbia University

In evidenzaRonald Davis at Columbia University

All the following studies have been made mainly thanks to private funding. Please, consider a donation to the Open Medicine Foundation, in order to speed up the research. See this page for how to donate to OMF.


OMF Scientific Advisory Board Director Ronald W. Davis, PhD, has just delivered a speech about ME/CFS at the Albert Einstein College of Medicine at Columbia University in New York. In what follows you find several screenshots that I have collected during the lecture, accompanied by a very short description. I imagine that a video will be soon made available but in the meantime let’s take a look at these slides.

75282243_10219023699079864_6188297319411089408_n.jpg

Indoleproprionate is reduced in ME/CFS patients. This molecule is not produced by our own metabolism, it comes from a bacterium of the gut (Clostridium sporogenes) which is low in patients. It has a neuroprotectant effect. Indoleproprionate is currently used in some clinical trials for other diseases and it might be available in the next future as a drug.

Hydroxyproline is high and this is believed to indicate collagen degradation. Ron Davis talked about the case of a ME/CFS patient who turned out to have a problem in the craniocervical junction which was fixed by surgery. Is there a link between high hydroxyproline and abnormalities of the joints (the neck among them) that some patients seem to have?

77328406_10219023493914735_6596293726195154944_n.jpg

The increase in electrical impedance in blood samples (as measured by the so-called Nanoneedle device) only happens when cells from ME/CFS patients are incubated with plasma from patients. When these same cells (white blood cells) are incubated with plasma from healthy donors, the impedance is normal. For an introduction to this experiment, click here. The published work is here.

74662392_10219023559036363_1916794623750045696_n-e1574360979550.jpg

The Nanoneedle study has been extended with 20 more patients and 20 more controls.  This device can be used for drug screening, other than for diagnosis.

76786420_10219023602557451_3286861926721650688_n.jpg

The peptide called Copaxone, now used in Multiple Sclerosis, seems to work in reducing the impedance in the nanoneedle device (click on the images to enlarge). Suramin also has some effect (on the right). It doesn’t seem as good as Copaxone though, to me…

SS-31 is an experimental drug for the mitochondrial membrane. It does work when used in the nanoneedle device! (click on the images to enlarge).

Nailbed capillaroscopy could be a new instrument for ME/CFS diagnosis. Inexpensive and already in use in hospitals.

75614142_10219023595157266_6718752052782039040_n.jpg

No new or known pathogen has been found in patients, so far. This project is still in progress. It is updated as new technologies for pathogen hunting become available.

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All the severe patients have at least one defective copy of the IDO2 gene. The same applies to 46 additional ME/CFS patients that have been recently tested for this gene. This is a common genetic problem in the general population, but it is ubiquitous in these patients. And a statistically significant difference is thus present between ME/CFS patients and healthy controls. This discovery has lead to the development of the metabolic trap hypothesis, which has been recently published (here). For an introduction, read this blog post of mine. They are planning to test the metabolic trap hypothesis in vivo using cellular cultures!

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Patients have high mercury (maybe from fish in their diet) and low selenium in hair. Low selenium can reduce the conversion of T4 to T3 in the liver. Low T3 might be a cause of fatigue. High uranium was also detected!

78209825_10219023656558801_2413390815666634752_n-e1574360171978.jpg


All these studies have been made mainly thanks to private funding. Please, consider a donation to the Open Medicine Foundation, in order to speed up the research. See this page for how to donate to OMF.


 

 

Maximum of a normal random vector

In evidenzaMaximum of a normal random vector

Ettore Majorana hadn’t got MATLAB

When Ettore Majorana first met Enrico Fermi, between the end of 1927 and the beginning of 1928, Fermi – who was already an acclaimed scientist in the field of nuclear physics – had just solved an ordinary differential equation of the second-order (whose solution is now commonly named the Thomas-Fermi function) – by numerical integration. It required a week of assiduous work for him to accomplish this task, with the aid of a hand calculator. Fermi showed the results (a table with several numbers) to Majorana, who was a 21 years old student of electrical engineering who had some vague idea of switching from the field of boring ways of providing electrical energy for boring human activities, to the quest for the intimate structure of matter, under the guide of Fermi, the brightest Italian scientific star of that period.

Majorana looked at the numerical table, as I wrote, and said nothing. After two days he came back to Fermi’s lab and compared his own results with the table made by Fermi: he concluded that Fermi didn’t make any mistake, and he decided that it could be worth working with him, so he switched from engineering to physics (Segrè E. 1995, page 69-70).

Only recently it has been possible to clarify what kind of approach to the equation Majorana had in those hours. It is worth mentioning that he not only solved the equation numerically, I guess in the same way Fermi did but without a hand calculator and in less than half the time; he also solved the equation in a semianalytic way, with a method that has the potential to be generalized to a whole family of differential equations and that has been published only 75 years later (Esposito S. 2002). This mathematical discovery has been possible only because the notes that Majorana wrote in those two days have been found and studied by Salvatore Esposito, with the help of other physicists.

I won’t mention here the merits that Majorana has in theoretical physics, mainly because I am very very far from understanding even a bit of his work. But as Erasmo Recami wrote in his biography of Majorana (R), a paper published by Majorana in 1932 about the relativistic theory of particles with arbitrary spin (Majorana E. 1932) contained a mathematical discovery that has been made independently in a series of papers by Russian mathematicians only in the years between 1948 and 1958, while the application to physics of that method – described by Majorana in 1932 – has been recognized only years later. The fame of Majorana has been constantly growing for the last decades.

The notes that Majorana took between 1927 and 1932 (in his early twenties) have been studied and published only in 2002 (Esposito S. et al. 2003). These are the notes in which the solution of the above-mentioned differential equation has been discovered, by the way. In these 500 pages, there are several brilliant calculations that span from electrical engineering to statistics, from advanced mathematical methods for physics to, of course, theoretical physics. In what follows I will go through what is probably the less difficult and important page among them, the one where Majorana presents an approximated expression for the maximum value of the largest of the components of a normal random vector. I have already written in this blog some notes about the multivariate normal distribution (R). But how can we find the maximum component of such a vector and how does it behave? Let’s assume that each component has a mean of zero and a standard deviation of one. Then we easily find  that the analytical expressions of the cumulative distribution function and of the density of the largest component (let’s say Z) of an m-dimensional random vector are

Eq 1.JPG

We can’t have an analytical expression for the integral, but it is relatively easy to use Simpson’s method (see the code at the end of this paragraph) to integrate these expressions and to plot their surfaces (figure 1).

surfaces.JPG
Figure 1. Density (left) and cumulative distribution function (right) for the maximum of the m components of a normal random vector. Numerical integration with MATLAB (by Paolo Maccallini).

Now, what about the maximum reached by the density of the largest among the m components? It is easy, again, using our code, to plot both the maximum and the time in which the maximum is reached, in function of m (figure 2, dotted lines). I have spent probably half an hour in writing the code that gives these results, but we usually forget how fortunate we are in having powerful computers on our desks. We forget that there was a time in which having an analytical solution was almost the only way to get mathematical work done. Now we will see how Majorana obtained the two functions in figure 2 (continuous line), in just a few passages (a few in his notes, much more in mine).

maximum.JPG
Figure 2.  The time in which Z reaches its largest value (left) and the largest value (right) in the function of the dimension of the normal random vector. The two curves in dotted lines are obtained through numerical integration (Simpson’s method) while the continuous lines are the function obtained by Majorana in an analytical way.
% file name = massimo_vettore_normale
% date of creation = 22/05/2019

clear all
delta = 0.01;
n(1) = 0.

for i=2:1:301;
  n(i) = delta + n(i-1);
  n_2(i) = - n(i);
end

for i=1:1:301
  f(i) = 0.39894228*( e^(  (-0.5)*( n(i)^2 )  ) );
end

for i=1:1:3
  sigma(1) = 0.;
  sigma(3) = sigma(1) + delta*( f(1) + ( 4*f(2) ) + f(3) )/3;
  sigma(2) = sigma(3)*0.5;
  for j=2:1:299
    sigma(j+2) = sigma(j) + delta*( f(j) + ( 4*f(j+1) ) + f(j+2) )/3;
  end  
end

for i=1:1:301
  F(i) =  0.5 + sigma(i);
  F_2(i) = 1-F(i);
end

for i=1:1:100;
  m(i) = i;
end

for i=1:1:301
  for j=1:1:100
    F_Z (i,j) = F(i)^j;
    F_Z_2 (i,j) = F_2(i)^j;
    f_Z (i,j) = 0.39894228*j*( F(i)^(j-1) )*( e^(  (-0.5)*( n(i)^2 )  ) );
    f_Z_2 (i,j) = 0.39894228*j*( F_2(i)^(j-1) )*( e^(  (-0.5)*( n(i)^2 )  ) );
  endfor
endfor

figure (1)
mesh(m(1:2:100),n(1:10:301),F_Z(1:10:301,1:2:100));
grid on
hold on
mesh(m(1:2:100),n_2(2:10:301),F_Z_2(2:10:301,1:2:100));
xlabel('m');
ylabel('t'); 
legend('F',"location","NORTHEAST");

figure (2)
mesh(m(1:2:100),n(1:10:301),f_Z(1:10:301,1:2:100));
grid on
hold on
mesh(m(1:2:100),n_2(2:10:301),f_Z_2(2:10:301,1:2:100));
xlabel('m');
ylabel('t'); 
legend('f',"location","NORTHEAST");

Asymptotic series

I have always been fascinated by integrals since I encountered them a lifetime ago. I can still remember the first time I learned the rule of integration by parts. I was caring for my mother who was dying. That night I was in the hospital with her, but she couldn’t feel my presence, she had a tumour in her brain and she was deteriorating. And yet I was not alone, because I had my book of mathematics and several problems to solve. But when my mind was hit by the disease for the first time, about a year later, and I lost the ability to solve problems, then real loneliness knocked at my door.

Now, why am I talking about the integration by parts? Well, I have discovered a few days ago, while studying Majorana’s notes, that integration by parts – well known by students to be a path towards recursive integrations that usually leads to nowhere – is in fact a method that can be useful for developing series that approximate a function for large values of x (remember that Taylor’s polynomials can approximate a function only for values of x that are close to a finite value x_0, so we can’t use them when x goes to ∞). Majorana used one such a series for the error function. He developed a general method, which I tried to understand for some time, without being able to actually get what he was talking about. His reasoning remained in the back of my mind for days, while I moved from Rome to Turin, where I delivered a speech about a paper on the measure of electric impedance in the blood of ME/CFS patients; and when I cried, some minutes later, looking at my drawings put on the screen of a cinema, Majorana was with me, with his silence trapped behind dark eyes. A couple of days later, I moved to a conference in London, searching for a cure that could perhaps allow my brain to be normal again and I talked with a huge scientist that once worked with James Watson. Majorana was there too, in that beautiful room (just a few metres from Parliament Square), sitting next to me. I could feel his disappointment, I knew that he would have found a cure, had he had the chance to examine that problem. Because as Fermi once said to Bruno Pontecorvo, “If a problem has been proposed, no one in the world can resolve it better than Majorana” (Esposito S. et al. 2003). Back in Rome, I gave up with the general method by Majorana and I found the way to calculate the series from another book. The first tip is to write the error function as follows:

Eq 2.JPG

Now by integrating by parts, one gets

Eq 3.JPG

But we can integrate by parts one other time, and we get

Eq 4.JPG

And we can go on and on with integration by parts. This algorithm leads to the series

Eq 5.JPG

whose main property is that the last addend is always smaller (in absolute value) than the previous one. And even though this series does not converge (it can be easily seen considering that the absolute value of its generic addend does not go to zero for k that goes to ∞, so the Cauchy’s criteria for convergence is not satisfied) it gives a good approximation for the error function. From this series, it is easy to calculate a series for the Gaussian function (which is what we are interested in):

Eq 6.JPG

A clever way to solve a transcendental equation if you don’t want to disturb Newton

Taking only the first two terms of the series, we have for the cumulative distribution function of Z the expression:

Eq 7.JPG

The further approximation on the right is interesting, I think that it comes from a well-known limit:

Eq 8.JPG

Now we can easily calculate the density of Z by deriving the cumulative distribution function:

Eq 9.JPG

With a further obvious approximation, we get:

Eq 10.JPG

In order to find the value of x in which this density reaches its largest value, we have to search for the value of x in which its derivative is zero. So we have to solve the following equation:

Eq 11.JPG

Which means that we have to solve the transcendental equation:

Eq 12.JPG

Majorana truncated the second member of the equation on the right and proposed as a solution the following one:

Eq 13.JPG

Then he substituted again this solution in the equation, in order to find ε:

Eq 14.JPG

With some further approximations, we have

Eq 15.JPG

So Majorana’s expression for the value of x in which the density of Z reaches its maximum value is

Eq 16.JPG

I have tried to solve the transcendental equation with Newton’s method (see the code below) and I found that Majorana’s solution is a very good one (as you can see from figure 3). Now, If we compare the approximation by Majorana with what I obtained using numerical integration at the beginning (figure 2) we see that Majorana found a very good solution, particularly for the value of x_M. Note: the transcendental equation that has been solved here seems the one whose solution is the Lambert W function, but it is not the same!

trascendental.JPG
Figure 3. The value of x in which the density of Z reaches its largest value. Majorana’s solution (continuous line) and a numerical solution obtained by means of Newton’s method (dotted line).
Vol II.JPG
Figure 4. From the original manuscript by Majorana, which can be found here (page 70). These are some of the passages that I have discussed here.
% file name = tangenti
% date of creation = 08/06/2019

clear all

x(1) = 1;            %the initial guess
for i=1:1:100
  m(i) = i;
end

for i=1:1:100
  for j = 2:1:1000
   f(j-1) = exp( 0.5*( x(j-1)^2 ) ) - ( m(i)/( x(j-1)*sqrt(2*pi) ) );
   f_p(j-1) = x(j-1)*exp( 0.5*( x(j-1)^2 ) ) + ( m(i)/( (x(j-1)^2)*sqrt(2*pi) ) );
   x(j) = x(j-1) - ( f(j-1)/f_p(j-1) );
   if ( abs(x(j)) < 0.001 )
    break;
   endif
   max_t (i) = x(j);
  endfor
endfor

% the aproximations by Majorana

for j=1:1:100
  max_t_M (j) = sqrt(log(j^2)) - ( log(sqrt(2*pi*log(j^2)))/sqrt(log(j^2)) );
endfor

% it plots the diagrams

plot(m(1:1:100),max_t (1:1:100),'.k','Linewidth', 1)
xlabel('m')
ylabel('time for maximum value')
grid on
hold on
plot(m(1:1:100),max_t_M (1:1:100),'-k','Linewidth', 1)

legend('numerical integration',"Majorana's approximation", "location", 'southeast')

Epilogue

From 1934 to 1938 Majorana continued his studies in a variety of different fields (from game theory to biology, from economy to quantistic electrodynamics), but he never published again (R), with the exception for a work on the symmetric theory of electrons and anti-electrons (Majorana E. 1937). But it has been concluded by biographers that the discoveries behind that work were made by Majorana about five years earlier and yet never shared with the scientific community until 1937 (Esposito S. et al. 2003). And in a spring day of the year 1938, while Mussolini was trying his best to impress the world with his facial expressions, Ettore became a subatomic particle: his coordinates in space and their derivatives with respect to time became indeterminate. Whether he had lived in a monastery in the south of Italy or he had helped the state of Uruguay in building its first nuclear reactor; whether he had seen the boundless landscapes of Argentina or the frozen depth of the abyss, I hope that he could have found, at last, what he was so desperately searching for.

He had given his contribution to humanity, so whatever his choice has been, his soul was already safe. And as I try to save my own soul, going back and forth from mathematics to biology, in order to find a cure, I can feel his presence. The eloquence of his silence trapped behind dark eyes can always be clearly heard if we put aside the noise of the outside world. And it tells us that Nature has a beautiful but elusive mathematical structure which can nevertheless be understood if we try very hard.

In the meanwhile, I write these short stories, like a mathematical proof of my own existence, in case I don’t have further chances to use my brain.

Until time catches me.

 

 

 

On the module of random vectors

In evidenzaOn the module of random vectors

“I have hopes of being able to achieve

something of value through my current studies or

with any new ideas that come in the future.”

J. F. Nash

The bridge over the Arno

In 1999 I was wandering in Pisa with a booklet in the pocket of a worn coat, too short for my frame. That coat was dark blue on the outside, green and red inside, with one mended sleeve and an austere cowl: I was much like a young monk, with his holy book (figure 1). I can remember neither its title nor the author, though. It was an introduction to statistical thermodynamics, with beautiful figures, a coloured cover, and less than 100 pages. It contained the work by Maxwell on the kinetic theory of ideal gasses, along with other material. I borrowed it from the University Library because I was fascinated by the way in which Maxwell was able to describe the properties of gasses with just a few hypotheses and some relatively easy mathematical passages. I felt that there was an enormous attraction in these methods, I realized with pleasure that math could give the power to completely understand and hold in hand physical systems and even, I started speculating, biological ones.

Analisi, coraggio, umiltà, ottimismo.jpg
Figure 1. A small self-portrait I drew in 1999 in an empty page of my pocket-sized French dictionary.

My second favourite composer back then was Gustav Mahler (the favourite one being Basil Poledouris): he represented my own way to classical music and I chose him because he wasn’t among the musicians my father and my brother shared a love for. I felt, during my teens, that I had to find my private space, and I met it one day on a used book stand: a cassette tape of Das Lied von The Erde, with a few sheets containing the translation to Italian of the songs. Mahler was born in 1860, a few weeks after Maxwell published his pivotal work about ideal gasses in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (R) (figure 2). But in 1999 I was repeatedly listening to a CD with a collection of songs sung by Edith Piaf and Charles Trenet, because I was studying French, and I was having a hard time with pronunciation. So, imagine a secular monk in his prime who listens to old French songs while keeping one hand on a book of statistical thermodynamics hidden in his pocket, wherever he goes, wandering in the streets of Pisa, a city which gave birth to Galileo Galilei. This seems a beautiful story, much like a dream, right? Wrong.

cover
Figure 2. Original cover of the journal in which Maxwell published his work on the kinetic theory of ideal gasses.

I had already started my struggle against the mysterious disease that would have completely erased my life in the following years. In the beginning, it had a relapsing-remitting course, so I could sometimes host the hope that I was recovering, only to find myself caught by the evil curse again. At the end of the year 1999, I was losing my mind, I knew that and I was also aware that my holy book couldn’t save me. I clearly remember one evening, I was walking on Ponte di Mezzo, a simple and elegant bridge above the Arno, and I felt that I couldn’t feel sorrow for the loss of my mind: I realized that not only the functions of my brain assigned to rational thinking were gone, but my feelings couldn’t function properly either. In fact, I noted without a true manifestation of desperation that I had lost my emotions. One day, after spending in vain about eleven hours on a single page of a textbook of invertebrate palaeontology, I accepted that I couldn’t read anymore, at least for the moment.

Had I known for sure that I wouldn’t have recovered in the following twenty years, I would have quite certainly taken my own life, jumping from a building; a fate that I have been thinking about almost every day ever since. I considered this possibility during the endless sequence of days in which there has been nothing other than the absence of my thoughts.

The distribution of velocities of an ideal gas and the one hundred years gap

In the already mentioned paper by Maxwell, he derived the probability density of the speed of the molecules of a gas, granted that the three components of the vector of speed are independent random variables (hyp. 1) and that they share the same density (hyp. 2), let’s say f. Moreover, the density of the speed has to be a function only of its module (hyp. 3). These three hypotheses together say that there is a function Φ such that

Cattura.PNG

This is a functional equation (i.e. an equation in which the unknown is a function) whose solution is not detailed in Maxwell’s work. But it can be easily solved moving to polar coordinates (see figure 3) and deriving with respect to θ both members (the second one gives naught since it depends only to the distance from the origin).

space of velocities.png
Figure 3. The space of velocities and its polar coordinates.

Another way to solve the functional equation is to use the method of Lagrange’s multipliers, searching for the extremes of the density of the velocity, when its module is fixed. In either case, we obtain the differential equation:

differential equation.png

which leads to the density for each component of the speed:

density.PNG

where σ can’t be determined using only the three hypotheses mentioned above. Considering then the well-known law of ideal gasses (pV=nRT) and an expression for p derived from the hypothesis that the collisions between the molecules of gas and the container are completely elastic, Maxwell was able to conclude that:

varianza.png

where m is the mass of the molecule of gas, T is the absolute temperature and K_B is the Boltzmann’s constant. It was 1860, Mahler’s mother was going to deliver in Kaliště, Charles Darwin had just released his masterpiece “On the origin of species”, forced to publish much earlier than what he had planned because of the letter he had received from Wallace, in which he described about the same theory Darwin had been working on for the previous 20 years. In the same point in time, Italy was completing its bloody process of unification, with the Mille expedition, led by Giuseppe Garibaldi.

But the functional equation I have mentioned at the beginning of this paragraph has brought with it a mystery for years, until 1976, when an employee at General Motors Corporation published a short note in the American Journal of Physics [R] in which he showed how Maxwell’s functional equation is, in fact, an example of the well known Cauchy’s functional equation:

Cauchy.png

In order to prove that, you just have to consider the following definition:

Nash.png

given that

Nash2.png

The name of the mathematician who made this observation is David H. Nash, and he has the merit of finding something new in one of the most known equation of physics, an equation that is mentioned in every book of thermodynamics, an equation that has been considered by millions of students in more than a century. It was 1976, my mother was pregnant with my brother; Alma, Gustav Mahler’s wife, had died about ten years before.

sphere.PNG
Figure 4. A sphere with a radius of z. The integrals of the density of within this sphere has the obvious meaning of the repartition function of Z.

Module of random vectors

Once Maxwell found the density of probability for each component of the speed of the molecules of an ideal gas, he searched for the density of the module of the speed. There is a relatively simple way of doing that. With the following notation

notazione.png

we have that the repartition function of Z is given by the integrals of the density of X within the sphere in figure 4. We have:

repartition function.PNG

The second expression is the same as above but in polar coordinates. Then we can obtain the density of Z by derivation of the repartition function. And this method can be extended to an m-dimensional space. This was the method used by Maxwell in his paper. And yet, there is another way to obtain the expression of the module of a random vector: I have explored it in the last months, during the rare hours in which I could function. By the way, only in the Summer of 2007 I was able to study the kinetic theory by Maxwell, eight years after I borrowed the holy book. Such a waste.

The hard way towards the density of the module of a random vector

When a student starts studying statistics, she encounters a list of densities: the normal distribution, the gamma distribution, the exponential distribution etc. Then there are several derived distributions that arise when you operate sums, roots extractions etc. on random variables. In particular, if f_X is the densty of X and Y = X², then we have

Quadrato.PNG

On the other hand, if Y = √X we have

Radice.PNG

Another important result that we have to consider is that given

linera combination.PNG

then

linear combination 2.PNG

By using these results I have been able to find that the expression of the density for  the module of an m-dimensional random vector is:

module.PNG

In particular, for m = 3 we have

module 2.PNG

Bessel surface.PNG
Figure 5. A plot of the modified Bessel function.

The case of normal random vectors: the modified Bessel function

In particular, if the random vector has dimension 3 and its components are normal random variables with the same expected value and variance, we have that the density of its module is given by

modulo normale.PNG

where I_0 is the modified Bessel function, which is one solution of the differential equation:

Bessel equation.PNG

whose name is modified Bessel equation. The integral expression of the modified Bessel function is:

Bessel function.PNG

I have coded a script in Matlab which integrates numerically this function (available here for download) which plots the surface in figure 5 and also gives the following table of values for this function.

Bessel table.PNG

The following is the flowchart of the script I coded.

flux chart.png

The case of normal random vectors with a naught expected value: the upper incomplete gamma function

If we consider random variables with an average that is zero (this is the case with the components of speed in ideal gasses), then the density is given by

Chi 3 scaled.PNG

which is a Chi distribution with 3 degrees of freedom, scaled with a scale parameter given by s = 1/σ. In the expression of the repartition function, ϒ is the lower incomplete gamma function, which is defined as follows:

incomplete gamma.PNG

I have written a code for its numerical integration (available here for download), the output of which is in figure 6.

lower incomlete.PNG
Figure 6. Lower incomplete gamma function.
maxwell surface.PNG
Figure 7. Maxwell-Boltzmann distribution for N2 in function of temperature and speed.
maxwell surface 2.PNG
Figure 8. Maxwell-Boltzmann distribution for Helium, Neon, Argon, at room temperature.

Conclusion, twenty years later

The density of the module of the velocity of the molecules of an ideal gas is, in fact, a scaled Chi distribution with 3 degrees of freedom, and it is given by

maxwell.PNG

It can be numerically integrated with the following script I made for Octave/Matlab, which gives the plot in figure 7. Another similar script gives the plot in figure 8. These plots represent the Maxwell-Boltzmann distribution, the centre of the holy book that an unfortunate boy was carrying in his pocket, all alone, some twenty years ago. He could have easily died by his own hand in one of the several thousand days of mental and physical disability that he had to face alone. Instead, he has survived. Had it been only for finding the Maxwell-Boltzmann distribution following another path, it would have been worth it. But he has found much more, including a bright girl, the magnificent next stage of evolution of human beings.

% file name = legge_Maxwell-Boltzmann_2

% date of creation = 22/02/2019

% it plots the density and the distribution function for the

% Maxwell-Boltzmann distribution considered as a function of temperature

% and speed

clear all

% we define some parameters

K_B = 1.381*10^(-23)                                                 % Boltzmann's constant

m = 4.65*10^(-26)                                                     % mass of the molecule N_2

% we define the array of temperature from 0° C to 1250° C

T (1) = 273.15

for i = 2:1:1250

T (i) = T (i-1) + 1.;

endfor

% it defines f_gamma in 3/2

f_gamma = sqrt(pi)/2.

% delta of integration

delta = 1.0

% it defines the array for the abscissa

z (1) = 0.;

for i = 2:1:2500

z (i) = z(i-1)+delta;

endfor

% it defines the density

for j = 1:1:1250

% it defines a constant

c = ( m/(K_B*T(j)) );

for i = 1:1:2500

f (j,i) = ( c^1.5 )*sqrt(2./pi)*( z(i)^2. )*( e^( -0.5*c*z(i)^2. ) );

endfor

% it calculates the ripartition function for He

F (j,1) = 0.;

F (j,3) = F (j,1) + delta*( f(j,1) + ( 4*f(j,2) ) + f(j,3) )/3;

F (j,2) = F (j,3)*0.5;

for k=2:1:2500-2

F (j,k+2) = F (j,k)+delta*( f(j,k)+( 4*f(j,k+1) )+f(j,k+2) )/3;

end

endfor

% It plots f and F

figure (1)

mesh(z(1:100:2500), T(1:100:1250), f(1:100:1250,1:100:2500));

legend('Density',"location","NORTHEAST");

xlabel('speed (m/s)');

ylabel('temperature (Kelvin)');

grid on

figure (2)

mesh(z(1:100:2500), T(1:100:1250), F(1:100:1250,1:100:2500));

legend('Probability',"location","NORTHEAST");

xlabel('speed (m/s)');

ylabel('temperature (Kelvin)');

grid on

 

Brain Normalization with SPM12

Brain Normalization with SPM12

Introduction

Brain normalization is the rigid rotation and/or deformation of a 3D brain scan so that it can match a template brain. This is a necessary step for the analysis of brain magnetic resonance data (whether it is morphological or functional) as well as of brain PET data. It allows introducing a set of spatial coordinates such that each triplet (x,y,z) identifies the same anatomical region both in the brain we are studying and in the template (Mandal P.K. et al. 2012). So, for instance, once the normalization has been performed on an fMRI of the brain of a patient and on a set of fMRIs from a suited control group, we can compare the BOLD activation of each anatomical region of the patient’s brain with the activation of the corresponding anatomical region of the control group.

Mathematical notes

This part can be skipped, it is not necessary to read these passages for the understanding of the following paragraphs. If we assume that P is a point of the brain before normalization and we call S(P) its position after the normalization, we can consider the vectorial function:

formula 1

which gives the new position of each point of the brain, after normalization. If P_0 is a point of the brain before the normalization, then we can write:

formula 2

and remembering the expression of the differential of a vectorial function, we have

formula 3

With a few passages we can write:

formula 4

From the above formula, we realize that in order to define the configuration of the brain after normalization, we have to define, for each point P, a set of 12 parameters. Of these parameters, 6 describe the rigid movement and can be considered the same for each point. The other 6 (the coefficients of the matrix) are those that describe the change of shape and size and, in general, they are different for each point. The job of SPM is to calculate these parameters.

Brain atlases

There are several templates (also called brain atlases) that have been developed across the years. The very first one was published in 1957 (Talairach J. et al. 1957). The same researcher then participated in building one of the most used brain atlas ever, based on a single female brain, published in 1988 (Talairach J. et Tournoux P. 1988). Another widely used template is the so-called MNI-152. It was built adapting the 3D MRI brain scans of 152 healthy individuals to the Talairach and Tournoux template. The adaptation was achieved using both a rigid roto-translation and a deformation (Maintz J.B. et Viergever M.A. 1988). The second step is required for overcoming the problem of differences in brain shape and dimension that we encounter within the human population.

Limitations

Available brain atlases have some limitations. One of them being the fact that despite diseased brains are the most widely studied, they are also the most difficult to register to a template built from healthy individuals, because of usually marked differences in shape and/or size. This is true for instance for brains of patients with Alzheimer’s disease (Mandal P.K. et al. 2012). Another important limitation is that registration algorithms perform poorly for the brainstem (particularly for pons and medulla) (Napadow V. et al. 2006). This might have represented a problem for the study of diseases where a possible involvement of the brainstem is suspected, like for instance ME/CFS (VanElzakker M. et al. 2019).

SPM12

The SPM software package is one of the most widely used instruments for the analysis of functional brain imaging data (web page). It is freely available for download, but it requires that you have a MatLab copy in your computer. Those who don’t have a MatLab license can request and install a standalone version of SPM12 by following the instructions of this page.

Importing a DICOM file

Once you have installed SPM12 in your computer, the first step in order to register a brain is to convert the format the series of images are written in, to a format that SPM12 can read. MRI images are usually in .dcm format (DICOM) while SPM12 reads .nii files. In order to do that, click DICOM import (figure below, on the left, red ellipse), then click on DICOM files (on the right, red ellipse), then select your .dcm file and click DONE (below, red ellipse). If you then click DISPLAY (blue ellipse, left) you will see your MRI scan in another window (see next paragraph). A video tutorial on these operations is available here.

DICOM import

Setting the origin

To start the normalization process, it is highly recommended to set manually the origin of the coordinates. If this is done properly, the registration will not only take less time but, even more importantly, the chances of a successful normalization will increase. The origin is set at the level of the anterior commissure (figure below). To find this anatomical structure, you can follow this video. Once you have put the cross on the right place in the sagittal, coronal and axial windows, just click SET ORIGIN (red ellipse) and then save your work clicking REORIENT (blue ellipse).

Origin

Normalization estimate

In this step, SPM12 calculates the set of distortions that have to be applied to the source brain to adapt it to the template in MNI space. On the main menu select NORMALIZE (ESTIMATE) (figure, on the left, red). This will open the batch editor where you are asked to load the subject you want to apply normalization to (figure, right, red). You have also a set of estimation options (blue), that we leave as they are.  Then you click the RUN button, the arrow on the top of the batch editor.

Normalization

At this point, your PC will perform a set of calculations that will require a few minutes. At the end of this process, a new .nii file will be saved in the spm12 folder. This is the set of distortions that will allow your subject’s brain to be registered to the template.

Normalization writing

Now click on NORMALIZE (WRITE) on the main menu. The batch editor will then ask you for the deformation field, which is the file generated in the previous step, and for the images to write, which is the scan of your subject (figure below). Select them, then press the RUN button on the batch editor. A new .nii file will be written in the spm12 folder. This is the normalized brain!

Normalization 2

In the next figure, you have the normalized brain on the left and the initial scan of the same subject on the right. As you can see, there is an overall change of shape.

Normalization 3

Anatomical areas

Now that we have normalized our brain in the MNI space, we can easily find anatomical regions within its sections. We can, for instance, load the normalized brain with MRIcron and overlay a template with Brodmann’s areas highlighted in various colours (figure below).

ws886245-0003-00001-000001-01.

I cannot die

I cannot die

For most ME/CFS patients (about two thirds), the disease has an oscillating course, with some periods of improvements followed by worsening of symptoms. Some of them can even experience recoveries, only to find themselves trapped again, weeks or months later (Stoothoff J et al. 2017), (Chu L. et al. 2019). Some anecdotes suggest that there might be a correlation with seasons, with improvements in summer, but there are no systematic surveys on that, to my knowledge.

As for me, in the last 20 years of pitiful combat with this monster, I experienced some substantial short-lived improvements, mainly during the core of summer. At the very beginning of the disease, I also recovered for a year. It was the year 2001, I was 21 and that year has been the only period of normality in my whole adult life. I spent these 12 months studying desperately and what I am as a person is mainly due to what I learned back then. I had already been very sick for about two years and when I recovered, it was as if I were born again. It was a second chance and I was determined to do all right from day one. I decided what was really important to me and I devoted myself to my goal: learning quantitative methods to use in engineering and – one day – in biology.

When darkness caught me again, I was, among other things, reviewing all the main theorems of calculus (particularly those about differential equations) with my new skills and I remember thinking that I was becoming good at developing my own proofs. I had become good at thinking and so, I reasoned, I could finally start my life! But in a few weeks, my mind faded away, and there was nothing I could do to keep a grip to all my beloved notes and books. They became mute and closed as monolithic gravestones. I remember clearly that along with this severe and abrupt cognitive decline, I developed also orthostatic intolerance, even though I hadn’t a name for it back then. But I couldn’t keep sitting, and I didn’t know why. I was forced to lay as if the gravitational acceleration had suddenly increased. My brain had changed to a lifeless stone, and so did my body.

From that very moment, my only thought has been how could I go back to my books and my calculations. And this still is my first thought, when I wake up in the morning. After almost 20 years.

I have experienced some short improvements in these years, during which I had to learn again how to study, how to do calculations, how to code. I never went back to what I was, though. And my brain is ageing, of course, as anyone else’s brain does. But in these short periods of miraculous come back I experience a rare sense of joy (along with anger and fear). Something that you can experience only if you have been facing death.

I was born and I died dozens of times in the last 20 years, and this gives me the perception that, in fact, I cannot die: I feel as if I were immortal and I had lived for a thousand years while at the same time still being in my twenties, since I have no experience of life.

In fact, I lived only when I crossed these short bridges from one abyss to the following one.

 

Immunosignature analysis of a ME/CFS patient. Part 1: viruses

Immunosignature analysis of a ME/CFS patient. Part 1: viruses

The purpose of the following analysis is to search for the viral epitopes that elicited – in a ME/CFS patient – IgGs against a set of 6 peptides, determined thanks to an array of 150.000 random peptides of 16 amino acids each. These peptides were used as query sequences in a BLAST search against viral proteins. No human virus was found. Three phages of bacterial human pathogens were identified, belonging to the classes Actinobacteria and γ-Proteobacteria. One of these bacteria, Serratia marcescens, was identified in a similar study on 21 ME/CFS cases.  

1. The quest for a pathogen

Scientists have been speculating about an infectious aetiology of ME/CFS for decades, without never being able to link the disease to a specific pathogen. The idea that the disease might be triggered and/or maintained by an infection is due to the observation that for most of the patients the onset occurs after an infectious illness (Chu, L. et al. 2019). It has also been observed that after a major infection (whether parasitic, viral or bacterial) about 11% of the population develops ME/CFS (Mørch K et al. 2013), (Hickie I. et al. 2006).

In recent years, the advent of new technologies for pathogen hunting has given renewed impulse to the search for ongoing infection in this patient population. A 2018 study, investigating the genetic profile of peripheral blood for prokaryotic and eukaryotic organisms reported that most of the ME/CFS patients have DNA belonging to the eukaryotic genera Perkinsus and Spumella and to the prokaryotic class β-proteobacteria (alone or in combination) and that these organisms are statistically more present in patients than in controls (Ellis J.E. et al. 2018). Nevertheless, a previous metagenomic analysis of plasma by another group revealed no difference in the content of genetic material from bacteria and viruses between ME/CFS patients and healthy controls (Miller R.R. et al. 2016). Moreover, metagenomic analysis pursued in various samples from ME/CFS patients by both Stanford University and Columbia University has come empty (data not published, R, R).

2. Immunological methods

Another way of investigating the presence of current and/or past infections that might be specific of this patient population is to extract the information contained in the adaptive immune response. This can be made in several ways, each of them having their own limits. One way would be to collect the repertoire of T cell receptors (TCRs) of each patient and see if they have been elicited by some particular microorganism. This is a very complex and time-consuming method that has been developed in recent years and that I have described in details going through all the recent meaningful publications (R). The main limitation of this method is that, surprisingly, TCRs are not specific for a single epitope (Mason DA 1998), (Birnbaum ME et al. 2014), so their analysis is unlikely to reveal what agent selected them. On the other hand, the advantage of this method is that T cell epitopes are linear ones, so they are extremely suited for BLAST searches against protein databases. An attempt at applying this method to ME/CFS is currently underway: it initially gave encouraging results (R), then rejected by further analysis.

Another possible avenue for having access to the information registered by adaptive immunity is to investigate the repertoire of antibodies. The use of a collection of thousands of short random peptides coated to a plate has been recently proposed as an efficient way to study the response of B cells to cancer (Stafford P. et al. 2014), infections (Navalkar K.A. et al. 2014), and immunization (Legutki JB et al. 2010). This same method has been applied to ME/CFS patients and it has shown the potential of identifying an immunosignature that can differentiate patients from controls (Singh S. et al. 2016), (Günther O.P. et al. 2019). But what about the antigens eliciting that antibody profile? Given a set of peptides one’s antibodies react to, a possible solution for interpreting the data is to use these peptides as query sequences in a BLAST search against proteins from all the microorganisms known to infect humans. This has been done for ME/CFS, and the analysis led to several matches among proteins from bacteria, viruses, endogenous retroviruses and even human proteins (in fact, both this method and the one previously described can detect autoimmunity as well) (Singh S. et al. 2016).  There are several problems with this approach, though. First of all, the number of random peptides usually used in these arrays is not representative of the variety of possible epitopes of the same length present in nature. If we consider the paper by Günther O.P. and colleagues, for instance, they used an array of about 10^5 random peptides with a length of 12 amino acids each, with the number of all the possible peptides of the same length being  20^12 ∼ 4·10^15. This means that many potential epitopes one has antibodies to are not represented in the array. Another important limitation is that B cell epitopes are mainly conformational ones, which means that they are assembled by the folding of the proteins they belong to (Morris, 2007), the consequence of this being that the subset of random peptides one’s serum react to are in fact linear epitopes that mimic conformational ones (they are often called mimotopes) (Legutki JB et al. 2010). This means that a BLAST search of these peptides against a library of proteins from pathogens can lead to completely misleading results.

Recently an array of overlapping peptides that cover the proteins for many know viruses has been successfully used for the study of acute flaccid myelitis (AFM). This technology, called VirScan, has succeeded in linking AFM to enteroviruses where metagenomic of the cerebrospinal fluid has failed (Shubert R.D. et al. 2019). This kind of approach is probably better than the one employing arrays of random peptides, for pathogen hunting. The reason being that a set of only 150.000 random peptides is unlikely to collect a significant amount of B cell epitopes from viruses, bacteria etc. Random peptides are more suited for the establishment of immunosignatures.

3. My own analysis

I have recently got access to the results of a study I was enrolled in two years ago. My serum was diluted and applied to an array of 150.000 peptides with a length of 16 random amino acids (plus four amino acids used to link the peptides to the plate). Residues Threonine (T), Isoleucine (I) and Cysteine (C) were not included in the synthesis of peptides. Anti-human-IgG Ab was employed as a secondary antibody. The set of peptides my IgGs reacted to has been filtered with several criteria, one of them being subtracting the immune response common to healthy controls, to have an immune signature that differentiates me from healthy controls. The end result of this process is the set of the following six peptides.

1 ALHHRHVGLRVQYDSG
2 ALHRHRVGPQLQSSGS
3 ALHRRQRVLSPVLGAS
4 ALHRVLSEQDPQLVLS
5 ALHVRVLSQKRPLQLG
6 ALHLHRHVLESQVNSL

Table 1. My immunosignature, as detected by an array of 150.000 random peptides 20-amino-acid long, four of which are used for fixing them to the plate and are not included here.

The purpose of the following analysis is to search for the viral epitopes that elicited this immune response. To overcome the limitations enumerated at the end of the previous paragraph I have decided to search within the database of viral proteins for exact matches of the length of 7 amino acids. Why this choice? A survey of a set of validated B cell epitopes found that the average B cell epitope has a linear stretch of 5 amino acids (Kringelum, et al., 2013); according to another similar work, the average linear epitope within a conformational one has a length of 4-7 amino acids (Andersen, et al., 2006). To filter the matches and to reduce the number of matches due to chance, I opted for the upper limit of this length. I excluded longer matches to limit the number of mimotopes for conformational epitopes. Moreover, I decided to look only for perfect matches (excluding the possibility of gaps and substitutions) so to simplify the analysis. It is worth mentioning that a study of cross-reactive peptides performed for previous work (Maccallini P. et al. 2018) led me to the conclusion that cross-reactive 7-amino-acid long peptides might often have 100% identity.

Sample
Figure 1. For each match, the matching protein and the organism it belongs to are reported. The protein ID has a link to the NCBI protein database, while the name of the organism has a link to the NCBI taxonomy browser. The host of the microorganism is also indicated, as well as its habitat, with links to further information.

4. Results

Table 2 is a collection of the matches I found with the method described above. You can look at figure 1 to see how to read the table.

ALHHRHVGLRVQYDSG (102_1_F_viruses)
9-LRVQYDS-15
QDP64279.1(29-35)
Prokaryotic dsDNA virus sp.
Archea, Ocean
8-GLRVQYD-14
AYV76690.1(358-364)
Terrestrivirus sp
Amoeba, forest soil
ALHRHRVGPQLQSSGS (102_2_F_viruses)
2-LHRHRVG-8
YP_009619965.1(63-69)
Stenotrophomonas phage vB_SmaS_DLP_5
Stenotrophomonas maltophilia (HP)
ALHRRQRVLSPVLGAS (102_3_F_viruses)
8-VLSPVLG-14
QDB71078.1 (24-30)
Serratia phage Moabite
Serratia marcescens (HP)
ALHRVLSEQDPQLVLS (102_4_F_viruses)
7-SEQDPQL-13
BAR30981.1 (151-157)
uncultured Mediterranean phage uvMED
Archea and Bacteria, Med. sea
ALHLHRHVLESQVNSL (102_6_F_viruses)
2-LHLHRHV-8
YP_009119106.1 (510-516)
Pandoravirus inopinatum
Acanthamoeba
4-LHRHVLE-10
ASZ74651.1 (61-67)
Mycobacterium phage Phabba
Mycobacterium smegmatis mc²155 (HP)

Table 2. Collection of the matches for the BLAST search of my unique set of peptides against viral proteins (taxid 10239). HP: human pathogen. See figure 1 for how to read the table.

5. Discussion

The are no human viruses detected by this search. There are some bacteriophages and three of them have as hosts bacteria that are known to be human pathogens. Bacteriophages (also known as phages) are viruses that use the metabolic machinery of prokaryotic organisms to replicate (figure 2). It is well known that bacteriophages can elicit specific antibodies in humans: circulating IgGs to naturally occurring bacteriophages have been detected (Dąbrowska K. et al. 2014) as well as specific antibodies to phages injected for medical or experimental reasons (Shearer WT et al. 2001), as reviewed here: (Jonas D. Van Belleghem et al. 2019). According to these observations, one might expect that when a person is infected by a bacterium, this subject will develop antibodies not only to the bacterium itself but also to its phages.

phage
Figure 2. Half of all viruses have an almost regular icosahedral shape, but several phages present an irregular icosahedral shape, with a prolate capsid (Luque and Reguera 2013). On the left a wrong representation of a phage. It is wrong because the capsid has 24 faces, instead of 20. On the right, the representation of a regular icosahedron made by Leonardo Da Vinci for De Divina Proportione, a mathematical book by Luca Pacioli.

If that is the case, we can use our data in table 2 to infer a possible exposure of our patient to the following bacterial pathogens: Stenotrophomonas maltophilia (HP), Serratia marcescens (HP), Mycobacterium smegmatis mc²155 (HP). In brackets, there are links to research about the pathogenicity for humans of each species. M. smegmatis belongs to the class Actinobacteria, while S. maltophila and S. marcescens are included in the class γ-Proteobacteria.

Interesting enough, Serratia marcescens was identified as one of the possible bacterial triggers for the immunosignature of a group of 21 ME/CFS patients, in a study that employed an array of 125.000 random peptides (Singh S. et al. 2016). This bacterium accounts for rare nosocomial infections of the respiratory tract, the urinary tract, surgical wounds and soft tissues. Meningitis caused by Serratia marcescens has been reported in the pediatric population (Ashish Khanna et al. 2013).

6. Next step

The next step will be to perform a similar BLAST search against bacterial proteins to see, among other things,  if I can find matches with the six bacteria identified by the present analysis. A further step will be to pursue an analogous study for eukaryotic microorganisms and for human proteins (in search for autoantibodies).

 

A leap of faith

A leap of faith

During last summer, I’ve pursued a lot of things. I delivered a speech in Turin, after the screening of the documentary Unrest, about the OMF-funded research on the use of the measure of blood impedance as a possible biomarker for ME/CFS (video, blog post, fig. 1, fig. 2).

62214141_2457107490968513_5354538085860245504_o
Figure 1. Group photo after the screening of Unrest in Turin, Italy, with the organizer of the event (Caterina Zingale, second from the right) and representatives of two Italian ME/CFS associations.
64982079_10217790422048709_8954534129534238720_o.jpg
Figure 2. Question time after the screening of Unrest. On the screen, a drawing of mine.

Then I flew to London to attend the Invest in ME conference, the annual scientific meeting that gathers researchers from all over the world who shared their latest work about ME/CFS. There I met Linda Tannenbaum, the CEO of the Open Medicine Foundation, whom I had the pleasure to encounter for the first time about a year before in Italy, and I introduced myself to Ronald Davis (fig. 3), the world-famous geneticists turned ME-researcher because of his son’s illness. I presented to him some possible conclusions that can be driven from the experimental results of his study on the electrical impedance of the blood of ME/CFS patients, with the use of an electrical model for the blood sample (R, paragraph 6).

61662643_10217615785202897_4802586779179810816_o.jpg
Figure 3. Talking to Ron Davis about a possible explanation for the increase in electrical impedance in the blood of ME/CFS patients in London, during the last Invest in ME conference (blog post).

In London, I was able to visit the National Gallery and while I was passing by all these artistic treasures without being able to really absorb them, to get an enduring impression that I could bring with me forever, I decided to sit down and to copy one of these masterpieces (I can’t draw for most of the time, and when I improve for a few weeks in summer, I usually have to carefully choose where to put my energies). I sat probably beside one of the least important portraits collected in the museum (Portrait of a young man, Andrea del Sarto, figure below) and I started copying it with a pen. When I finished, the museum was closing, so that I missed all the works by Van Gogh, among many other things.

We were at the beginning of June, I was experiencing my summer improvement, a sort of substantial mitigation of my illness that happens every other summer, on average. But because of these travels, I elicited a two-month worsening of symptoms, during which I had to stop again any mental and physical activity: I just lay down and waited. At the beginning of August, I started thinking and functioning again and I almost immediately decided to quit what was my current project (a 600-page handbook of statistics that I commenced in 2017) and I started studying mathematical modelling of enzymatic reactions (figures 4 and 5).

Cattura.JPG
Figure 4. The plot of the reversible Michaelis-Menten equation for ribulose-phosphate 3-epimerase. The intersection of the surface with the plane is the state of equilibrium of the reaction (the rate of production of X5P is the same as the rate of its transformation into Ru5P) (R).
sa.png
Figure 5. The plot of the concentrations of S, P, ES and E for the transformation of L-tryptophan into N-formylkynurenine by IDO-2. This is a semianalytical solution that I found using the approximation of the Lambert function. I also pursued numerical solutions, of course (R).

I knew that these reactions were described by ordinary differential equations and that I could solve them numerically with the methods that I studied just before I got sick, about 18 years ago. I was interested in the metabolic trap theory by Robert Phair, an OMF-funded researcher. So I downloaded a chapter of one of the most known books of biochemistry and a thesis by a Turkish mathematician on metabolic pathways simulation and I started my journey, working on the floor (I have orthostatic intolerance even when I get better in Summer, so I can’t use a desk, figure 6). I ended up learning the rudiments of this kind of analysis, also thanks to a book by Herbert Sauro and to some suggestions by dr. Phair himself! Some of the notes I wrote in August are collected here.

69099435_10218252327836065_3701013412482908160_o.jpg
Figure 6. In August I was studying mathematical modelling of metabolic pathways, sitting on the floor: because of my orthostatic intolerance, I can’t sit for long, especially when I need to think. You can see the book by Sauro opened in the foreground, the thesis by a Turkish mathematician (red cover), and the handbook of Octave (blue/red cover).

At the beginning of September, I was absorbed by the problem of how to study the behaviour of the steady states of tryptophan metabolism in serotoninergic neurons of midbrain as the parameters of the system change. This kind of analysis is called bifurcation theory and I literally fell in love with it… In figure 6 you can also see a drawing: I was drawing a picture I have been thinking about for the last 20 years. It is a long story, suffice it to say that in 1999, just before my mind faded away for 18 months, I started studying the anatomy of a man who carries a heavy weight on his back (see below). That was my first attempt of communicating what was happening to me, of describing my disease.

Only recently I considered to not represent the weight, which is a more appropriate solution since this is a mysterious disease with no known cause, and I made a draft (the one in figure 2) that I then used as a starting point for the drawing below. I finished this new drawing at the beginning of September, in a motel room of San José, in California, just in time for donating it to Ronald Davis (figures below) when I moved to the US to attend the third Community Symposium at Stanford (see here). In California, many surprising things happened: I met again Linda Tannenbaum and Ronald Davis, and yes, I encountered also Robert Phair! But this is another story…

resilience 2

69836282_10218351956686724_1046414613542862848_o

In the following pictures, you can see how the drawing evolved. Notably, the figure in the centre changed his face and some part of his anatomy. The three figures are meant to be a representation of the same figure from three different points of view. It is more like a project for a sculpture, a monument that is much deserved by these patients.

At Stanford, I had the chance to be face to face with one of my preferred sculptures ever: The Thinker, by Rodin, in both its version: the model moulded first, on the top of The Gates of Hell, and the big one (crafted later), now considered the iconic symbol of Philosophy, but likely originally meant to be a metaphor for creative thinking (I say that because the original sculpture included in The Gates of Hell is a representation of the Italian poet Dante Alighieri, depicted in the act of imagining his poem).

At the end of September, my mind started fading away again. I knew that would have happened, even though I had an irrational hope that this year would have been different. At that point, I was in Italy and I asked some friends to help me organize a trip to the southern hemisphere, in order to live another summer. It required more time than I would have hoped. I am going to leave from Italy only tomorrow. My goal: Argentina. I have been able to do something, at a highly reduced speed, in October, though. I have developed a model for solar radiation at sea level, in function of the day of the year, of the latitude, and of the distance from the Sun (I have considered the actual elliptic orbit of our planet). The main problem has been the modelling of absorption and of diffusion of radiant energy from our star by the atmosphere, but I solved it. Part of these notes are here, but I want to self-publish the end product, so I keep the rest to myself. In that period, I was also able to find the exact solution of the improper integral known as the Stefan-Boltzmann law, something I tried to do in the summer of 2008, in vain, in one of my recovery-like periods. In figure 6 you can see one of the results of my model for solar radiation: the monochromatic emissive power at sea level in function of the day of the year, for the city of Buenos Aires.

emissive power.png
Figure 7. The monochromatic emissive power of the sun at noon, at sea level, at a latitude of -32° N, in function of the wavelength and of the day of the year. Note that the vertical axis is expressed in W on microns multiplied by square meters.

My intention was to use that model to choose the perfect place where to move in order to have environmental conditions that closely resemble the ones that we have in Rome from June to September (the period in which my improvements happen). I also wanted to quantitatively study the effect of both infrared radiation and ultraviolet radiation on my biology. There are several interesting observations that can be made, but we will discuss these subjects another time, also because I had to quit this analysis given my cognitive deterioration. The video below is a byproduct of the geometric analysis that I had to pursue in order to build my model for solar radiation at sea level.

Dawn and dusk at a latitude of 42 degrees north, during three years of the silent rolling of the Earth on its silken ellipse. Three years of adventures, suffering, joy and death.

So, by November my mind was completely gone and my physical condition (namely orthostatic intolerance and fatigue) had worsened a lot. This year I have been able to try amphetamines: I had to go from Rome to Switzerland to buy them (they are restricted drugs that can’t be sold in Italy and can’t be shipped to Italy either). One night I felt good enough to take a train to Milan and then to take another transport to the drug store. And back. I managed to do the travel but I pushed my body too far and I had to spend the following month in bed, 22 hours a day, with an even worse mental deterioration. It is like having a brain injury. Amphetamines have been useless in my case, despite two studies on their potential beneficial effect in ME/CFS.

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Right now, I am collecting all the books and the papers that I need with me in Argentina (figure above), in case I will improve enough to study again. But what am I going to work on?

  1. I want to finish my model of solar radiation, with some notes on the effect of infrared radiation, ultraviolet radiation and length of the day on the immune system. There is a mathematical model published recently that links the length of the day to the power of the innate immune system, and I want to write a code that calculates the relative activity of the innate immunity in function of latitude and day of the year. I would like to self- publish it as a booklet.
  2. I want to finish my handbook of statistics.
  3. I need to correct a paper submitted for publication (it has been accepted, but some corrections have been required).
  4. I want to deepen my understanding of the bifurcation theory for metabolic pathways and to continue studying tryptophan metabolism with this new knowledge.
  5. I want to complete my work on autoantibodies in ME/CFS (see this blog post) and to submit it to a journal. I have been working on that for a while, inventing new methods for the quantitive study of autoimmunity by molecular mimicry.

Should I improve again in Argentina, several avenues can be followed in order to understand the reason why summer causes this amelioration in my own case. I have many ideas and I’ll hopefully write about that in the future. Of course, I also want to read all the new research papers I have missed in the last months. I will bring with me my handbook of anatomy for artists because I hope to be able to draw again, and I won’t miss this opportunity to leave some other handcrafted images behind me for posterity, that can’t care less, obviously! I would really like to finish the drawing below because I feel that in this draft I have found a truly elegant (and mechanically correct) solution for the hip joint of a female robot.QR-A_000026

Now I am useless, my mind doesn’t work and I am housebound. I can’t read, I can’t draw, I can’t do calculations, I can’t do coding, I can’t cook… This has been the quality of my life for most of the last 20 years. This is a huge waste: I would have used these years to perform beautiful and useful calculations and to pursue art. I would really make people understand how tragic this disease is in its cognitive symptoms, what we lose because of it. This is, in fact, the reason behind this blog post: I wanted to give an idea of what I can do when I feel better, and of what I would have done if there had been a cure.

I have lost most of my adult life, but I will never accept to waste a day without fighting back.

Historia magistra vitae?

Historia magistra vitae?

What is happening with the CCI-hypothesis in the ME/CFS community closely resembles what happened in Italy (mainly, but not only) with the CCVI-hypothesis of Multiple Sclerosis (MS). There was this new avenue, completely unexpected and very fascinating (to me, at least), that linked MS to a defect in the venous system of the neck, named Chronic cerebrospinal venous insufficiency (CCVI) by the Italian researcher Paolo Zamboni [1]. Several MS patients underwent surgery to correct one or more veins of the neck and described themselves as cured of MS thanks to this surgery. Among them also a prominent patient advocate, Pavarotti’s wife, who gave enormous publicity to this kind of technique [2].

The diagnosis of CCVI was somehow subjective, and only CCVI-literate doctors could do it properly. The same applied for the surgery. Several surgeons in private practice started doing the surgery on MS patients, earning a lot of money in a very short period of time.

Does this seem familiar?

After a decade and several well-designed studies, no correlation between CCVI and MS has been demonstrated [3], [4].

I am not saying that there is no correlation between CCI and ME/CFS. We don’t know yet. I personally find interesting these new hypotheses about the effect of abnormal mechanical strains on the functioning of the brainstem and the possible link to ME/CFS-like symptoms and I am trying to study this new field (see this blog-post), among all the other hypotheses about the aetiology of ME/CFS.

What I would like to point out with this post is that it is perfectly possible that several patients improve with this kind of surgery even in the absence of any link between CCI and ME/CFS. This is a weird (and fascinating) phenomenon that we have already seen in other diseases. It always has the same pattern: a somehow subjective diagnosis that only a few physicians can do, a surgery or a drug that many physicians are warning against, a huge amount of patients who say that they have recovered after the intervention.

C’è del marcio a Bologna?

C’è del marcio a Bologna?

I disabili possono costituire una fonte di reddito cospicua per chi lavora nell’ambito della assistenza a queste persone. Per le disabilità riconosciute, infatti, il sistema sanitario offre (giustamente) dispositivi costosi e personale con le più varie mansioni. Ma in molti casi appalta questo genere di servizi.

Come funziona un appalto? Si redige un capitolato di gara, che elenca i requisiti che i candidati devono possedere per poter partecipare alla gara. Questo capitolato viene reso pubblico solo al momento del bando. Il vincitore sarà l’ente che, tra i candidati, meglio soddisfa i requisiti del capitolato di gara.

Un anno fa, dopo una indagine di vari mesi, la Guardia di Finanza di Bologna ha iscritto nel registro degli indagati quattro dipendenti Ausl e due rappresentanti della Associazione Italiana Assistenza Spastici (AIAS Bologna) con l’accusa di aver truccato un appalto. Infatti il file del capitolato di gara sarebbe stato corretto più volte – prima ancora del bando – dai rappresentanti di AIAS Bologna, in modo da cucirsi addosso il concorso e assicurarsi la vittoria [1], [2].

Si parla di due milioni e 130 mila euro circa, in tre anni, con possibile rinnovo per altri tre anni. Si tratta della gara per il Centro Regionale Ausili (che include anche il Centro Ausili Tecnologici) e per il Centro Adattamento Ambiente Domestico. La gara è stata sospesa [3], [4].

Dopo ulteriori accertamenti, il procedimento contro uno dei dipendenti Ausl è stato archiviato, ma per il resto l’indagine è ancora in piedi, che io sappia [5].

Io stesso sono il beneficiario dei servizi di una di queste aziende appaltatrici, che assistono le persone con disabilità. E non potrei fare a meno di questi servizi, non sopravviverei.

Proprio per questo mi infastidisce non poco venire a sapere di tali illeciti (presunti, fino a che il procedimento giudiziario non termina il suo corso). E perché sono spiacevoli e pericolosi questi illeciti? Perché in fondo suggeriscono che le disabilità più remunerative sono quelle a cui si presta più attenzione e a cui si offrono più servizi.