|A image of asoulism or disbelief in the existence of souls, with a play on the similarly sounding words "soul" (of humans) and "sole" (of shoes).|
The term "asoulism" is a relatively new term, seemingly coined by David Weisman (2010) as follows: 
“I wish there were a term in the English language that honestly captures the idea that all we experience is due to brain function. ‘Materialism’ comes close, but is laden with excess metaphysical baggage. The philosopher John Searle coined ‘biological naturalism’ as a mind-body theory within philosophy, and that comes very close. ‘Asoulism’ is more modest: a simple disbelief in the existence of souls based on evidence.”— David Weisman (2010), “The Experience of a Unified Mind and the Possibility of an Everlasting Soul are Connected and there is Scant Evidence to Support the Existence of Either” 
Examples of avowed asoulists include: Jean Meslier (1729), Julien la Mettrie (1745), Francois Broussais (c.1820), Thomas Edison (1824), Kurt Bell (2011), Patrick Fergus (2014), and Libb Thims (2015).
Soulless | Bag of chemicals
See main: Soulless bag of chemicals
In 2008, in Breaking Bad, third episode, “… And the Bag’s in the River”, season one, the show opens to Professor White cleaning up the human that had been dissolved in hydrofluoric acid (shown adjacent; lower left), amid which his mind reflects back to, it seems, his college days, and dialogue with a woman, possibly his lover at the time, as shown adjacent right (main image), wherein she reads off the percentage of elements of a human, in moles, to him and he writes them down on a chalkboard, as follows:
|An illustrated diagram, with the CHNOPS elements enlarged, of the "soul" scene from the third episode, season one, of Breaking Bad, wherein Walter White and a woman list the elements of a human and speculate on the "soul" as the thing missing from the list.|
Hydrogen (H) 63%
Oxygen (O) 26%
Carbon (C) 9%
Nitrogen (N) 1.25%
Calcium (Ca) 0.25%
Phosphorus (P) 0.19%
Sodium (Na) 0.04%
Iron (Fe) 0.00004%
at the end of which he comments how he thinks that something is missing, to which the woman replies “the soul”?
In 2011, secular thinker Kurt Bell, an article/blog titled “Soulless Bag of Chemicals”, opened to the following: 
“On my better days I regard my wife and daughter as ‘soulless bags of chemicals.’ To be fair though, I think the same of myself and all other forms of life. Souls are supernatural things beyond perception or measure, worthy only of suspended belief pending some real evidence or the giving up of the tenets of science. Chemicals we are…despite the clever animation of flesh and musings of mind brought about by the electricity of life. So why do we think more of ourselves? How are we so offended by these facts?”
Bell then followed this up with a YouTube vlog about how his use of the controversial phrase about how he thought of his wife and his daughter as “soulless bags of chemicals”, which he said he heard somewhere, borrowed it, and liked it, but that it sparked a heated reaction and “got under the skin” of many people, no pun intended.
Thereafter, into the 2010s, the phrase began to take root, whether arisen independently or via the "heard somewhere" method, or by possibly by Bell (or others). The following are a few example usages of the phrase:
“Think about atheism for a second. As an atheist, you must believe that you are the result of the purely mindless, random chance interaction of particles over an immensely long period of time—the classic monkey typing Shakespeare scenario. (I know that the origin of species involves natural selection—however what created our universe with natural laws, which make life possible? Blind chance.) Being a soulless bag of chemicals created by unguided, meaningless random chance you of course have no free will. You are merely a zombie acting automatically according to the chemicals swishing around in your brain. I don't find this too plausible for many reasons.”— Jacob Stein (2012), “Does Egyptian History Contradict the Torah”, Jan 10 
“If Darwin was right, man is nothing more than a soulless bag of chemicals or equally soulless hominid and there is no ultimate fixed meaning or higher purpose to life except what we choose.”— Linda Kimball (2013), “Our Only Defense Against the Destructive ‘Me’ Generation & Antichrist Utopianists”, Aug 20 
The term "soulless bag of chemicals", as of 2014, has yet to make it into GoogleBooks, and seems to presently reside in the spontaneous colloquial vernacular of the blogosphere, videos, and forums.
One tricky issue that compounds the jump to asoulism, is that of "moralism" connected to the theory of the soul, namely that if one denies the soul, one also, assumedly, must therefore, be an amoralist, i.e. deny the existence of "morals" (or justice).
A second, less tricky, issue that compounds the jump to asoulism, is that of "mortalism" connected to the theory of the soul, namely that if one denies the soul, one also, assumedly, must therefore be an mortalist, i.e. deny the existence of "immortality" (or afterlife), or variants thereof.
Partial | Asoulism
There exists a middle range of atheists who deny the existence of god but retain a semblance of soul theory; the following is one notable example:
“There was no wit, no shadow, no nuance, no multi-sidedness, no art, no pleasure or erotic play to be found in its pages. It denied god, argued that all religions were created out of fear, ignorance, and anthropism; that souls did not outlive the body; that the world was determined by strict laws.”— Denis Diderot (c.1769), private complaints on reading Baron d’Holbach’s The System of Nature 
Another example is the 1899 evolution-based neuro-soulplasm theory of atheist-pantheist Ernst Haeckel. A few "skeptical agnostic asoulists" include: Napoleon Bonaparte (c.1815), Sigmund Freud (sort of: Yes/No; metaphor), and possibly David Weisman (2010).
The following are related quotes:
“What is the soul, but an empty word to which no idea corresponds?”— Julien la Mettrie (c.1747/48) 
“A soul? Give my watch to a savage, and he will think it has a soul. If I have a soul, then pigs and dogs also have souls”— Napoleon Bonaparte (1817), “To Gaspard Gourgaud”, St. Helana, Jan 28
“The church believes in things that do not exist, in ‘souls’; it believes in effects that do not exist, in ‘divine’ effects; it believes in states that do not exist, in ‘sin’, in ‘redemption’, in the salvation of the soul: it stays everywhere on the surface, at signs, gestures, word to which it gives an arbitrary meaning. It possesses a thoroughly thought-out method of psychological counterfeiting.”— Friedrich Nietzsche (1888), WP:394, Mar-Jun
“One of the proofs of the immortality of the soul is that myriads have believed it. They also believed the world was flat.”— Mark Twain (c.1902), Notebook (Ѻ)
“The naturalist now readily admits that plants have souls — or will-power — but he appropriates the soul as an energy of thermodynamics.”— Henry Adams (1910), per citation of Wilhelm Wundt 
“My mind is incapable of conceiving such a thing as a soul. I may be in error, and man may have a soul; but I simply do not believe it. What a soul may be is beyond my understanding.”— Thomas Edison (1924), “Do We Live Again?” 
“We need to demythologize persons by rooting out certain unfounded ideas from the perennial philosophy. Letting go of the belief in souls is a minimal requirement. In fact, desouling is the primary operation of the scientific image.”— Owen Flanagan (2002), The Problem of the Soul; cited by Julien Musolino (2015) in The Soul Fallacy (pg. 21)
“The luckiest thing that ever happened to me was that my father didn’t believe in god, and so he had no hang-ups about souls. I see ourselves as products of evolution, which itself is a great mystery.”— James Watson (2003), “Discover Dialogue: Reversing Bad Truths” 
“In making such a claim (‘soul does not exist’; ‘ether does not exist’) I am circumscribing legitimate concepts and claims from illegitimate concepts and claims. We do not then say that ‘souls are examples of nothingness’; to do so would be to understand …”— Brett Bourbron (2004), Finding a Replacement for the Soul 
“As a physicist who had never seen the hand of god in any observable phenomena, I view the scientific rejection of the soul as the ultimate validation of reductionist material monism.”— Victor Stenger (2005), “Forward” to The Soul Fallacy (pg. 9) by Julien Musolino
“Where do we come from? Do souls really exist? I can’t answer these questions, especially not at 6am.”— Miranda July (c.2011) (Ѻ)
“There's no soul; that’s garbage.”— Patrick Fergus (2014), response statement when Libb Thims told Pat that he was conducting a “What is the Soul Comprised of?” polling of 36 people; also picked as most-liked answer by vote, Sep 28 
● Amoralism | Moralism
● Geniuses on the soul
1. Weisman, David. (2010). “The Experience of a Unified Mind and the Possibility of an Everlasting Soul are Connected and there is Scant Evidence to Support the Existence of Either” (Ѻ) (Ѻ), Seed, Jul 3.
2. Edison, Thomas. (1924). “Do We Live Again?” (Ѻ), The Illustrated London News, May 3.
3. Stott, Rebecca. (2013). Darwin’s Ghosts: in Search of the First Evolutionists (Voltaire, pg. 157). A&C Black.
4. Fergus, Patrick. (2014). “What is the Soul Comprised Of?” [response by Fergus] (Poll A) [N=36] / Pick the best answer (Poll B) [N=16] [Fergus answer picked #1 answer by vote], poll conducted by Libb Thims, Sep 28.
5. (a) La Mettrie, Julien. (1745). Man a Machine and Man a Plant (soul, 52+ pgs). Hackett Publishing, 1994.
(b) Ball, Philip. (2011). Unnatural: the Heretical Idea of Making People (pgs. 97-99). Vintage Books.
6. Bourbon, Brett. (2004). Finding a Replacement for the Soul: Mind and Meaning in Literature and Philosophy (soul does not exist, pg. 214). Harvard University Press.
7. (a) Watson, James. (2003). “Discover Dialogue: Reversing Bad Truths”, Discover, Jul.
(b) Mills, David. (2006). Atheist Universe: the Thinking Person’s Answer to Christian Fundamentalism (pg. 225). Ulysses Press.
8. Adams, Henry. (1910). A Letter to American Teachers of History. Publisher.
● Musolino, Julien. (2015). “You Don’t Have a Soul: the Real Science that Debunks Superstitious Charlatans” (Ѻ), Salon, Jan 25.