THE MOTION OF THE SPHERICAL COW: NUMERICAL 
SIMULATIONS OF THE DECELERATION OF A MULTI-ATOM 
SYSTEM BY A MOTION THROUGH A GASEOUS 
ENVIRONMENT 
 
 
 
 
 
 
 
by 
JULIA ABDOULLAEVA  
 
 
 
 
 
 
 
 
 
A THESIS 
 
Presented to the Department of Chemistry and Biochemistry  
and the Robert D. Clark Honors College  
in partial fulfillment of the requirements for the degree of  
Bachelor of Science 
 
May 2024 
 
 
 
An Abstract of the Thesis of 
Julia Abdoullaeva for the degree of Bachelor of Science 
in the Department of Chemistry and Biochemistry to be taken May 2024 
 
 
Title:  The Motion of the Spherical Cow: Numerical Simulations of the Deceleration of a Multi-
Atom System by a Motion Through a Gaseous Environment 
 
 
 
 
Approved:          Jeffrey Cina Ph.D.        _ 
               Primary Thesis Advisor 
 
Particle collision theory is applied in experimental settings such as mass spectroscopy and other 
particle fragmentation techniques to gain insight into observed behaviors. The energetic 
interactions that occur between large macromolecules and ions during a collision are complex 
and require several analytical tools and techniques to fully understand. The research described in 
this thesis raises the question of whether it is possible to develop a numerical simulation that 
simplifies this collision model and allows us to form a general understanding of particle 
interactions during a collision. Several computational methods are implemented in the 
development of this program including Monte Carlo, a binning procedure, and a double rolling 
procedure. The numerical simulation will aim to calculate an appropriate background gas particle 
distribution and relevant trajectories such as the monomer’s center of mass velocity.  
 
 
 
 
2 
 
 
Acknowledgements 
 
First I would like to thank my primary thesis advisor, Dr. Jeffrey Cina, for inviting me to work 
on this thought provoking research question and for guiding my learning. This research has 
played a huge role in sparking my passion for theoretical physical chemistry, and for inspiring 
me to pursue graduate studies. I am very thankful for all of the time and effort that was spent 
educating me. I would like to thank Dr. Rebecca Altman for being my CHC representative on 
this thesis. I would like to thank the Prell Lab at the University of Oregon for providing us with 
the experimental basis for the numerical simulation. I would like to thank the Pathway Oregon 
Program and the Summit Scholarship Program for funding my undergraduate education.  
  
3 
 
 
Table of Contents 
Introduction 6 
Computational Methods 11 
Monte Carlo Method 11 
Bins Procedure 12 
Double-Roll Procedure 14 
Results and Discussion 17 
Conclusions and Future Directions 25 
Appendix 26 
Bibliography 38 
 
 
  
4 
 
 
List of Figures 
 
Figure 1. Simulation setup ............................................................................................................ 10 
Figure 2. Distribution of particle velocities .................................................................................. 17 
Figure 3. Bin Distribution ............................................................................................................. 18  
Figure 4. Modified Bin Distribution ............................................................................................. 18  
Figure 5. Distribution of Velocities in the nth Bin ....................................................................... 19 
Figure 6. Analytical Convergence to the Boltzmann Distribution at 1000 velocities .................. 20 
Figure 7. Analytical Convergence to the Boltzmann Distribution at 10,000 velocities ............... 21 
Figure 8. Analytical Convergence to the Boltzmann Distribution at 100,000 velocities ............. 22 
Figure 9. Analytical Convergence to the Boltzmann Distribution at 1,000,000 velocities. ......... 23 
Figure 10. Monomer center of mass velocity ............................................................................... 24  
 
 
5 
 
 
Introduction 
Numerical simulations in physical chemistry are essential tools for gaining a better 
understanding of experimental outcomes. The numerical simulation described in this thesis was 
developed in order to create a simple model of protein behavior in a gaseous environment. The 
original experiment was developed by the Prell Lab at the University of Oregon to determine 
mass fragmentation patterns of a protein structure.1 Electrospray ionization (ESI) is an analytical 
technique in chemistry that aims to transfer a folded protein into the gas phase through ion 
mobility. 1 Ion mobility is a research technique that aims to promote separation of ionized 
molecules, as well as their identification. This technique is particularly relevant because of its 
ability to preserve elements of the initial structure along with the global size measurement.  1 The 
global size measurement is a way to describe the size of a complex using the mass to charge 
ratio. The application of these two techniques in conjucture is referred to as native ion mobility-
mass spectrometry (IM-MS). 1 IM-MS is relevant to experiments in bioanalytical chemistry, as it 
can be used as a technique to determine structural biology through a perturbation of the complex 
using gas-phase activation along with IM-MS. 1 Gas-phase activation is a technique used to 
fragment gas molecules through collisions with molecular ions. In the experiment of interest, 
IM-MS was implemented alongside gas-phase activation methods to determine the quaternary 
and tertiary structures of a protein complex. 1 IM-MS begins by introducing ions into an 
isothermal and isobaric gas chamber that is influenced by an electrical field. 1 As collisions 
occur, the technique promotes the degeneration of the protein complex to measure overall 
fragmentation patterns of the protein. 1 The numerical simulation takes foundational elements 
from the processes that occur within the IM-MS experiment to develop features of the 
6 
 
 
simulation. Although we aren’t using the results from the simulation to make experimental 
insights, it is important to use realistic principles to develop a simple model.  
Collision-induced unfolding (CIU) as well as collision-induced dissociation (CID) are 
combined in tandem to induce collisions between gas phase particles and the protein complex.  1 
During the applications of CIU and CID, protein ions are accelerated to reach a high kinetic 
energy, and several thousand protein ions are forced to collide with the neutral gas atom. The 
collisions will eventually lead to the heating of the protein ions. 1 Once the protein ions surpass 
the maximum threshold for structure stability, the protein will begin dissociation or unfolding.  1 
When a protein complex undergoes CID, it can provide information about the complex 
stoichiometry, as well as the identity of subunits. 1 Application of CIU allows experimenters to 
determine other important features such as the overall structure, binding patterns, and 
specificities of the ligand binding site. 1  
 In 1989, an article titled ‘Theory of Collisional Activation of Macromolecules. Impulsive 
Collisions of Organic Ions’ was published by chemists at the University of Oslo describing 
computational methods for collisional activation. 2 The experiment that is initially described 
involves the guidance of ions towards a gaseous cell. 1 Energy transfers that occur between the 
ions and the gas cell result in ion fragmentation due to a technique referred to as collisional 
activation (CA). 2 Collisional activation is performed by positioning a collision cell directly 
between two mass analyzers. 2 The first mass analyzer (MS1) will transmit the selected ion, and 
the second mass analyzer (MS2) will scan and record the fragmented ions. 2 Collisional 
activation can be described using gas-phase kinetics, and the following model is used to show the 
time-dependent process of collisional activation and the resultant reaction that occurs because of 
the kinetic energy transfer. 2 This model denotes molecules A and B in various stages of 
7 
 
 
collisional activation. The ionic complex AB+ undergoes a collision with a gas atom, G, to form 
the energized complex (A-B+)*. 
 
(1) 𝐴𝐵 + 𝐺 → (𝐴 − 𝐵 )∗ + 𝐺∗ 
𝑘(𝐸), 𝑃(𝐸)
(2) (𝐴 − 𝐵 )
∗   𝐴 + 𝐵 
→
Once the collisions have occurred, the energy distribution of the ions are referred to as the 
function P(E). 2 The distribution of P(E) is dependent on the transfer of energy between the ion 
beam and the internal degrees of freedom of the gas cell. 2 It is concluded that the rate of 
fragmentation during the experiment is dependent upon the distribution of P(E) and the rate 
constants k(E). 2 The ion fragmentation rate is determined by the unimolecular rate constants 
k(E) as well as the internal energy distribution P(E). 2 The theory of impulsive collision transfer 
(ICT) is a mathematical model used to describe energy transfers due to particle collisions in the 
experiment. 2 ICT defines the macromolecular ion as a sum of atoms that have masses m  2a,j.  The 
summation of ma,j is defined as the total mass of the ion m  2ion.  Prior to the collision and 
subsequent energy transfer, the macromolecular ion has a relative velocity in the lab frame of 
reference and is traveling on the defined x-axis within the mass spectrometer. 2 The lab frame of 
reference referes to the laboratory setting where the experiment of interest is performed. The 
kinetic energy and momentum of the ion are defined by equations 3 and 4. 2 
1 1
(3) 𝐸 = 𝑚 , 𝑣 = 𝑚 𝑣  2 2
(4) 𝑝 = 𝑚 , 𝑣 = 𝑚 𝑣 
In order to increase the quantity of fragmented ions, the amount of energy that is channeled into 
the system is increased by transferring a large amount of energy into the macromolecular ion.  2 
8 
 
 
The center of mass energy of the macromolecular ion can be expressed using the following 
equation. 2 This equation indicates the maximum energy that can be used for the transfer to the 
internal degrees of freedom of the system. 2  
(5) 𝑚𝐸 = 𝐸 
𝑚 + 𝑚
The foundational theoretical concepts from the impulsive collision transfer (ICT) model 
are the basis for the development of the numerical simulation. The numerical simulation 
simplifies the treatment from a macromolecular ion to a unit referred to as the “spherical cow”. 
The spherical cow is a concept in theoretical physics that is used to describe highly simplified 
models. In the numerical simulation, the particle is modeled as a mono-unit spherical cow that 
participates in linear, one-dimensional motion through a gaseous environment. As described by 
the ICT model, the monomer will experience frequent collisions via gas atoms that result in 
changes of the total energy and center of mass velocity of the monomer. In my simulations, the 
focus is on the decrease over time of the monomer’s center of mass velocity. An important 
feature of the numerical simulation that is modeled after the gas-phase activation procedure in 
IM-MS, is a theoretical distribution of gas particles. The gas particle distribution is modeled after 
the infamous Boltzmann distribution which can be written in the following form where E is the 
energy of the system, kB is the Boltzmann constant, and T is the temperature of the system. 3  
(6) 𝑃 ∝  𝑒  
The Boltzmann distribution is applied in later calculations, along with factors accounting for the 
relative velocity of the monomer and a gas atom. Generally, this distribution is used to determine 
the behavior of the gas particles and how they interact with the monomer.  
We can define two types of collisions that may occur during the simulation. A backside 
collision occurs when a gas particle hits the monomer from behind, this results in energy transfer 
9 
 
 
that alters the center of mass velocity. Similarly, a frontside collision is when the gas particle hits 
the monomer from the front. This will also result in an alteration of the center of mass velocity. 
Since the monomer is moving in one dimension, a backside collision will visually occur on the 
left side of the monomer, and a frontside collision will occur on the right side of the monomer. 
This can be visualized using the diagram modeled in Figure 1.These principles are used to 
develop the computational methods described below that can perform calculations and obtain 
results for further analysis.  
 
Figure 1. Simulation setup 
 
10 
 
 
Computational Methods 
Monte Carlo Method 
 A prevalent concept in computational chemistry that is often applied to large statistical 
ensembles is a method referred to as Monte Carlo. 4 Monte Carlo procedures have a wide range 
of applications in research such as macromolecular structures and polymers. The method uses 
conditional probability to determine the appropriate statistical distributions for a system of study.  
4 In the case of this simulation, a Monte Carlo method was developed in order to create a 
Boltzmann distribution of background gas particles that exist within an appropriate range of 
velocities. The appropriate range of velocities will be defined as any velocity between -vmax and 
+vmax where the variable vmax can be computed as follows where Tg and mg are the temperatures 
of the background gas and the mass of the background gas particles, respectively.  
2𝑇
(7) 𝑣 = 𝐿𝑜𝑔[1000] 
𝑚
My application of the Monte Carlo method was developed using the Mathematica software, and 
follows a procedure that is programmed to test each velocity. Randomized velocities are 
generated through a While/If loop structure where the initial step implements a RandomReal 
function. RandomReal is a Mathematica function that will generate a pseudorandom number 
between 0 and 1. The RandomReal function can be combined alongside other functions to 
generate an appropriate randomized number for the situation at hand. In the case of the 
Boltzmann distribution, we can combine the RandomReal function with a function that generates 
both negative and positive numbers, and multiply it by vmax to create an output that exists within 
the specified range. The program will then take the velocity, and generate a corresponding 
11 
 
 
probability using the Boltzmann distribution. The procedure will implement a condition for the 
calculated probability; if the probability is greater than a pseudorandom number generated by the 
RandomReal function, the condition is met and the gas particle velocity can be appended to a list 
of velocities. If the condition is not met, the program will restart the procedure until we reach the 
indicated number of gas particle velocities. The program is set to acquire one million gas particle 
velocities in order to have a complete understanding of the nature of this distribution.  
 
Bins Procedure 
 Once the Monte Carlo method has generated the correct amount of gas particle velocities, 
we then implemented a procedure to categorize each velocity into a corresponding bin. The 
program will generate 200 bins, each with specific criteria for binned velocities. The width of 
each bin is defined as twice the value of vmax divided by the number of bins. A Do loop structure 
is utilized to test each velocity and determine the correct bin placement using the Floor function. 
The Floor function takes a number as an input and outputs the greatest integer less than or equal 
to the input. In this procedure, the Floor function is used to take each gas particle velocity and 
output the number of the corresponding bin. The function is written as follows where vsb[[nn]] 
is the specified gas particle velocity, vmax is the maximum gas particle velocity, and dv is the 
width of each bin.  
(8) 𝑣𝑠𝑏 [𝑛𝑛] + 𝑣𝑏𝑖𝑛𝑠 = 𝐹𝑙𝑜𝑜𝑟 + 1 
𝑑𝑣
The Do loop will test each velocity using this procedure. The output of this function is the 
number of the corresponding bin that will be populated. It is important to note that the bin does 
not contain the numerical value of the velocity itself, instead it contains an integer which 
12 
 
 
represents the number of velocities that corresponds to that bin. Once every velocity is tested we 
obtain a rough representation of the Boltzmann distribution within the ranges of -vmax and +vmax. 
We can then take the rough distribution and create a modifed bin distribution by normalizing 
each value. This is accomplished by creating a new list of values that takes each bin and divides 
that value by the total number of elements in the original list of velocities. We can now 
implement a function that tests each modified bin for the middle value. We can define a function 
that calculates the middle of each bin using the following syntax where the variable n indicates 
the bin number.  
(9) 𝑑𝑣𝑚𝑛𝑏[𝑛 ] ≔  −𝑣 + + 𝑑𝑣(𝑛 − 1) 
2
This function is used in tandem with a normalized Boltzmann distribution function which is 
written as follows in equation 10; 
𝑚 ( [ ])
(10) 𝑃(𝑚𝑛𝑏[𝑛]) = 𝑒 𝑑𝑣 
2𝜋 ∗ 𝑇𝑔
The value for the middle of each bin is inputted into the normalized Boltzmann distribution, and 
the value obtained from the normalized bin distribution is placed into a list that when plotted is a 
representation of the distribution of velocities in each bin. We can then perform an analytical 
comparison between the original Boltzmann distribution and the normalized bin distribution to 
show a convergence to the Boltzmann distribution as we increasingly populate the velocity bins. 
This analytical convergence to the Boltzmann distribution is shown as we go from a distribution 
as small as 1000 velocities to the ideal range of 1,000,000 velocities. The convergence is an 
important factor to keep track of because we want to ensure the simulation is behaving properly. 
By performing the procedure at a wide range of velocities, we ensure that the simulation is 
actually converging to the ideal model.  
13 
 
 
Double-Roll Procedure 
 Now that we have a functioning distribution of gas particle velocities, we need to 
determine how the gas particles will interact with the monomer. The double roll procedure will 
determine when and what kind of collisions occur. There are two types of collisions that are 
relevant to the numerical simulation; a backside collision and a forward collision. This 
distinction is important to note because we will use slightly different probabilities and 
probability densities to describe the behavior of each type of collision. In the case of a backside 
collision we can define a probability for a collision between the monomer and a gas particle as 
follows where  is the linear density of the background gas particles, dt is the change in time,  
is a variable defined as the mass of the gas particles divided by twice the temperature of the gas, 
and v1 is the monomer velocity.  
1 𝑣
(11) 𝑝 (𝑣 , ∞) = 𝛿𝑡𝜌{ 𝑒 − 𝑒𝑟𝑓𝑐[√𝛼𝑣 ]} 
2√𝜋𝛼 2
At the beginning of the timestep, the simulation uses the RandomReal function to select a 
number between 0 and 1. In the case where the randomized number is greater than the 
probability function, there is no collision that occurs during that timestep, and the procedure 
restarts. In the case where a collision does occur, we need to determine what type of collision 
occurs and ensure that the gas particle exists within an appropriate range of velocities. This part 
of the procedure is performed through a guess-and-select procedure using the probability density 
for backside collisions. The probability density is defined as follows where vg is the gas particle 
velocity.   
𝛼
(12) 𝑤 𝑣 , 𝑣 = 𝛿𝑡 𝑣 − 𝑣 𝑈𝑛𝑖𝑡𝑆𝑡𝑒𝑝[𝑣 − 𝑣 ]𝜌 𝑒  
𝜋
 
14 
 
 
The probability density will take its maximum value when vg is equal to vg,bar. vg,bar is defined as 
follows in equation 13;  
1 2
(13) 𝑣 , = 𝑣 + 𝑣 +   2 𝛼
 
The following function will serve as a weighting function which accepts values between zero and 
unity. This can accept or reject values for vg within the range of -vg,max and vg,max.  
𝑤 (𝑣 , 𝑣 ) 𝑣 − 𝑣
(14) = 𝑈𝑛𝑖𝑡𝑆𝑡𝑒𝑝(𝑣 − 𝑣 ) 𝑒 ( , ) 
𝑤 (𝑣 , 𝑣 , ) 𝑣 , − 𝑣
 
After the weighting function determines the type of collision that occurs, we calculate what is 
referred to as the bump in velocity. This refers to the velocity changes that the monomer 
experiences as a result of frequent collisions, and is used to track the monomer’s center of mass 
velocity. The velocity bump is calculated at the end of each successfully completed timestep, and 
can be expressed as follows where mg is the mass of one gas particle, m is the mass of the 
monomer, and vmf[[jj]] is the previous monomer velocity that is used for the updated 
calculation.   
2𝑚
(15) 𝑣 = (𝑣 − 𝑣𝑚𝑓 [𝑗𝑗] ) 
𝑚 + 𝑚
The velocity bump is added to the previous velocity to calculate the overall change in center of 
mass velocity. The probability for head on collisions is defined as follows.  
(16) 1 𝑣𝑝 (𝑣 , −∞) = 𝛿𝑡𝜌{ 𝑒 + 𝑒𝑟𝑓𝑐[−√𝛼𝑣 ]} 
2√𝜋𝛼 2
Head on collisions will only occur if the rolled number between zero and unity is less than the 
probability for head on collisions to occur. The corresponding probability density is defined as 
follows.  
15 
 
 
𝛼
(17) 𝑤 𝑣 , 𝑣 = 𝛿𝑡𝜌 𝑣 − 𝑣 𝑈𝑛𝑖𝑡𝑆𝑡𝑒𝑝(𝑣 − 𝑣 ) 𝑒  
𝜋
The probability density has a maximum value at the value vg,dbar which can be written using the 
following syntax.  
1 2
(18) 𝑣 , = 𝑣 − 𝑣 +  2 𝛼
The weighting funcztion for head on collisions, is written as follows, and is used to select gas 
particle velocities within the appropriate range for head on collisions.  
𝑤 (𝑣 , 𝑣 ) 𝑣 − 𝑣
(19) = 𝑈𝑛𝑖𝑡𝑆𝑡𝑒𝑝 𝑣 − 𝑣 𝑒 ( , ) 
𝑤 (𝑣 , 𝑣 , ) 𝑣 − 𝑣 ,
 
This part of the procedure will also calculate the bump of velocity after each successful timestep, 
and the updated velocity will be appended to the list of center of mass monomer velocities. Once 
the procedure has been performed the desired amount of times, we can plot the list of monomer 
velocities to view changes in the center of mass velocity throughout the simulation.  
16 
 
 
Results and Discussion 
The Monte Carlo procedure allows us to generate a distribution of gas particle velocities, 
as shown below in Figure 2. The velocity list, where the gas particle velocities are stored, has 
1000 list elements appended to it, each with a varying distribution of velocities ranging 
anywhere from -3 vg,max  d/t to 3 vg,max  d/t .  
Figure 2. Distribution of particle velocities 
 
 
As described earlier, each velocity is binned according to its numerical value, and is represented 
in Figure 3 as the unmodified bin distribution. The bin distribution is later modified as is shown 
in Figure 4, in order to create a normalized representation of the bin distribution for later 
comparisons with the Boltzmann distribution.  
17 
 
 
Figure 3. Bin Distribution  
Bin Distribution
20
15
10
5
0
0 50 100 150 200
Bin  
Figure 4. Modified Bin Distribution  
Modified Bin Distribution
0.020
0.015
0.010
0.005
0.000
0 50 100 150 200
Bin  
 
18 
 
Modified Number of Corresponding Velocities Number of Velocities
 
The probability distribution described in Figure 5, shows the probability of a gas particle 
velocity exisiting in the nth bin. The distribution is modeled after the Boltzmann distribution, we 
can see that the greatest probability for a gas particle existing in a bin, is in the center bin of the 
overall distribution. As you go farther from the center bin, the probability decreases and 
approaches zero. 
 
Figure 5. Distribution of Velocities in the nth Bin 
Distribution of Velocities in the nth Bin
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
0 50 100 150 200
Bin  
 
In Figures 6-9, we plot the modified bin distribution together with the probability 
distribution described in Figure 5 at varying numbers of velocities in order to show analytical 
convergence. We begin at 1000 velocities in Figure 6 and eventually reaching 1,000,000 
velocities in Figure 9. As the number of velocities is increased, we see that the modified bin 
distribution eventually begins to converge to the probability distribution. This indicates that the 
binning distribution closely follows a Boltzmann probability distribution, which is the desired 
19 
 
Probability of Existing in the Velocity of the Nth Bin
 
result for this computation. The Boltzmann distribution is the ideal model for the simulation 
because the Boltzmann distribution describes the behavior of an infinite collection of gas particle 
velocities. If the simulation was performed to obtain an infinite collection of gas particles, the 
distribution would ideally perfectly match a Boltzmann distribution.   
 
Figure 6. Analytical Convergence to the Boltzmann Distribution at 1000 velocities 
 
20 
 
 
Figure 7. Analytical Convergence to the Boltzmann Distribution at 10,000 velocities 
 
21 
 
 
Figure 8. Analytical Convergence to the Boltzmann Distribution at 100,000 velocities 
 
 
22 
 
 
Figure 9. Analytical Convergence to the Boltzmann Distribution at 1,000,000 velocities.  
 
The double-roll procedure described earlier allows us to plot the monomer’s center of 
mass velocity which is graphed in Figure 10. The monomer has an initial velocity of 5.0 d/t, and 
within a duration of 200,000 timesteps we see the monomer’s velocity slow down to 
approximately 3.5 d/t in a stepwise way. Although it is expected that the monomer’s center of 
mass velocity slows down, the stepwise pattern is unexpected. Initial predictions for the center of 
mass velocity is that it will slow down in a smooth pattern, but the results show that the velocity 
oscillates between a specific range of values and then rapidly drops down. For the purposes of 
this simulation, the current results are valid, but are open to further testing and development to 
investigate this pattern. 
 
 
23 
 
 
 
Figure 10. Monomer center of mass velocity 
 
Center of Mass Velocity
3.9
3.8
3.7
3.6
3.5
0 50 000 100 000 150 000 200 000
TimeStep  
 
 
  
24 
 
Monomer Velocity
 
Conclusions and Future Directions 
The results gained from this numerical simulation have allowed us to paint an initial 
picture of what occurs when a single-unit spherical cow interacts with a distribution of gas 
particle velocities. These initial results indicate that the Monte Carlo procedure and the binning 
procedure create a desired distribution of gas particle velocities that is modeled after a modified 
version of the Boltzmann distribution. We can also say that the monomer’s center of mass 
velocity slows down throughout the course of the simulation. The numerical simulation has 
several future goals that need to be achieved in order to gain a more conclusive picture of 
particle collisions in a mass spectroscopy experiment. The next step that needs to be taken is a 
simulation of two harmonically bound particles, and how they experience heating and cooling 
due to energy transfers from collisions with the gas particles. The end goal of this program is to 
simulate a spherical cow with many components, in order to get a more conclusive picture. 
 
 
 
  
25 
 
 
Appendix 
 
  
26 
 
 
 
  
27 
 
 
 
 
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29 
 
 
 
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Bibliography 
1. Donor, M.; Shepherd, S. O.; Prell, J. S. Rapid Determination of Activation Energies 
for Gas-Phase Protein Unfolding and Dissociation in a Q-IM-ToF Mass 
Spectrometer. Journal of the American Society for Mass Spectrometry 2020, 31 (3), 
602–610. https://doi.org/10.1021/jasms.9b00055. 
 
2. Einar Uggerud, and Peter J Derrick. “Theory of Collisional Activation of 
Macromolecules. Impulsive Collisions of Organic Ions.” The Journal of Physical 
Chemistry, vol. 95, no. 3, 1 Feb. 1991, pp. 1430– 
 
3. Leonard Kollender Nash. Elements of Statistical Thermodynamics. Mineola, New 
York, Dover Publ., Cop, 2006. 
4. Allen, Michael P, and Dominic J Tildesley. Computer Simulation of Liquids. 
Oxford,Oxford University Press, 2017.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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